


1 « Ifflilf if }({[{ 


l?s?l!l](ii 



■I 



I) 



m s m a m maaaaffl 









% 









\* 






-?> v « V<^ 



.A 















- A^ 






























^ 


J ni" 




-£> 






'^r. 


,#' 


- 




% 











^> ^- 















A^ 

- 



^" * 












*> '^. 









^V 






#% 

s 



X 0o x. 



^ 'V 



\s 






- . ^ 



v'\ 






C' 












lOo, 









% S^ 









\0 



"^ i i « 



. " 















V ' ^ 



<£, 



X A 






\ °*. 

























^ 


















A TREATISE 



ON 



HYDRAULICS. 



PROFESSOR OF CIVIL ENGINEERING IN LEHIGH UNIVERSITY, 



MANSFIELD MERR1MAN, 

; N\ 



FIFTH EDITION, REVISED AND ENLARGED. 

FIRST THOUSAND. 



NEW YORK: 
JOHN WILEY & SONS, 

53 East Tenth Street, 
1895. 



o^> ■■ 






Copyright, 1889, 

BY 

Mansfield Merriman. 



Dbummont> & Net/, 

Electrotypers, 
1 to 7 Hague Street. 

New York Printers, 

rk " 326 Pearl Street, 

New York. 



LC Control Number 




tmp96 025678 



CONTENTS 



Chapter I. Introduction . 

Art. i. Units of Measure 

2. Physical Properties of Water 

3. The Weight of Water 

4. Atmospheric Pressure 

5. Compressibility of Water 

6. The Acceleration of Gravity 

7. Numerical Computations 

Chapter II. Hydrostatics . 

Art. 8. Transmission of Pressures 
9. Head and Pressure 

10. Normal Pressure . , 

11. Pressure in a given Direction 

12. Centre of Pressure on Rectangles 

13. General Rule for Centre of Pressure 

14. Pressures on Opposite Sides of a Plane 

15. Masonry Dams 

16. Loss of Weight in Water 

17. Depth of Flotation . 

18. Stability of Flotation 

Chapter III. Theoretical Hydraulics 

Art. 19. Velocity and Discharge . 

20. Velocity of Flow from Orifices 

21. Orifices whose Plane is Horizontal 

22. Rectangular Vertical Orifices . 

23. Triangular Vertical Orifices 

24. Circular Vertical Orifices 

25. Influence of Velocity of Approach 

26. Flow under Pressure 

27. Pressure Head and Velocity Head 

28. Time of Emptying a Vessel 

29. Flow from a revolving vessel . 



PAGE 

I 

1 
3 
5 
7 
9 
10 
12 

14 
14 
15 

18 
20 
22 
24 
27 
28 
30 
31 
33 

36 

36 
37 
40 
42 
44 
45 
46 

49 

52 
56 
58 



IV COXTEXTS. 

PAGE 

Art. 30. The Path of a Jet 61 

31. The Energy of a Jet 64 

32. The Impulse and Reaction of a Jet ..... 66 

33. Absolute and Relative Velocities ...... 68 

Chapter IV. Flow through Orifices 71 

Art. 34. The Standard Orifice . . m 71 

35. The Coefficient of Contraction ...... 73 

36. The Coefficient of Velocity 74 

37. The Coefficient of Discharge 76 

38. Vertical Circular Orifices . . . . . .78 

39. Vertical Square Orifices 80 

40. Vertical Rectangular Orifices 82 

41. The Miner's Inch . . . . . . . .84 

42. Submerged Orifices 86 

43. Suppression of the Contraction ...... 87 

44. Orifices with Rounded Edges ....... 88 

45. Measurement of Water by Orifices ..... 89 

46. The Energy of the Discharge . ..... .92 

47. Discharge under a Variable Head ...... 94 

48. Emptying and Filling a Canal Lock 96 

Chapter V. Flow over Weirs - . 9 s 

Art. 49. Description of a Weir ........ 9^ 

50. The Hook Gauge 100 

51. Formulas for the Discharge . 102 

52. The Velocity of Approach ....... I0 5 

53. Weirs with End Contractions ....... 107 

54. Weirs without End Contractions ...... no 

55. Francis' Formulas 112 

56. Submerged Weirs . . . . . . . . .114 

57. Rounded and Wide Crests ....... ll 7 

58. Waste Weirs and Dams 119 

59. The Surface Curve ......... 122 

60. Triangular and Trapezoidal Weirs . . . . . 1 -4 

Chapter VI. Flow through Tubes 128 

Art. 61. The Standard Short Tube 12S 

62. Conical Converging Tubes 130 

63. Nozzles and Jets ......... 132 

64. Diverging and Compound Tubes ...... 137 

65. Inward Projecting Tubes . ...... 141 

66. Effective Head and Lost Head ...... 142 

67. Losses in the Standard Tube 145 



CONTENTS. V 

PAGE 

Art. 68. Loss due to Enlargement of Section 148 

69. Loss due to Contraction of Section . . . . .151 

70. Piezometers .......... 153 

71. The Venturi Water Meter , 158 

Chapter VII. Flow in Pipes 162 

Art. 72. Fundamental Ideas ......... 162 

73. Loss of Head at Entrance ....... 164 

74. Loss of Head in Friction ....... 166 

75. Other Losses of Head 170 

76. Formula for Velocity ........ 173 

77. Computation of Discharge ....... 175 

78. Computation of Diameter ....... 177 

79. Short Pipes .......... 179 

80. Long Pipes 181 

81. Relative Discharging Capacities ...... 182 

82. A Compound Pipe ......... 184 

83. Piezometer Measurements ....... 186 

84. The Hydraulic Gradient 189 

85. A Pipe with Nozzle ........ 193 

86. House-service Pipes ........ 196 

87. A Water Main . . .198 

88. A Main with Branches . 201 

89. Pumping through Pipes ........ 203 

90. Fire Hose .......... 207 

91. Lampe's Formula 208 

92. Very Small Pipes ......... 210 

Chapter VIII. Flow in Conduits and Canals . . . .212 

Art. 93. Definitions ., . . . . . . . . . 212 

94. Formula for Mean Velocity . . . . . . . 215 

95. Circular Conduits, full or half full 218 

96. Circular Conduits, partly full ....... 221 

97. Open Rectangular Conduits ....... 224 

98. Trapezoidal Sections ........ 227 

99. Horseshoe Conduits 231 

100. Lampe's Formula 232 

101. Kutter's Formula 233 

102. Sewers ........... 235 

103. Ditches and Canals ........ 239 

104. Losses of Head 242 

105. The Energy of the Flow . . ... . . . 245 

Chapter IX. Flow in Rivers 247 

Art. 106. Brooks and Rivers ......... 247 



VI CONTENTS. 

PAGJS 

Art. 107. Velocities in a Cross-section 249 

108. The Transporting Capacity of Currents . . . .251 

log. The Current Meter ........ 253 

no. Floats ........... 256 

in. Other Current Indicators . . . . . . . 258 

112. Gauging the Flow . . 260 

113. Gauging by Surface Velocities ...... 262 

114. Gauging by Sub-surface Velocities ...... 264 

115. Comparison of Methods ....... 266 

116. Variations in Velocity and Discharge 268 

117. Non-uniform Flow ........ 270 

ri 3. The Surface Curve 273 

119. Backwater 277 

Chapter X. Measurement of Water Power .... 283 

Art. 120. Theoretic and Effective Power 283 

I2i. Measurement of the Water 285 

122. Measurement of the Head 288 

123. Determination of Effective Power 290 

124. The Friction Brake, or Power Dynamometer . . . 292 

125. Test of a Small Motor ........ 295 

126. The Lowell and Holyoke Tests ...... 298 

127. Water Power of Rivers and of the Tides .... 302 

Chapter XI. Dynamic Pressure of Flowing Water . . 304 

Art. 128. Definitions and Principles 304 

129. Experiments on Impulse and Reaction ..... 307 

130. Surfaces at Rest ......... 310 

131. Curved Pipes and Channels . 313 

132. Immersed Bodies ......... 316 

133. Moving Vanes ......... 318 

134. Work derived from Moving Vanes ...... 322 

135. Revolving Vanes ......... 326 

136. Work derived from Revolving Vanes ..... 32S 

137. Revolving Tubes 332 

Chapter XII. Naval Hydromechanics 335 

Art. 138. General Principles ......... 335 

139. Frictional Resistances ........ 337 

140. Work required in Propulsion ....... 340 

141. The Jet Propeller . 341 

142. Paddle Wheels 343 

143. The Screw Propeller 345 

144. The Action of the Rudder „ 34 5 

145. Tides and Waves ......... 349 



CONTENTS. VI 1 

PAGE 

Chapter XIII. Water Wheels 352 

Art. 146. Conditions of High Efficiency. ...... 352 

147. Overshot Wheels ......... 354 

148. Breast Wheels . 35 8 

149. Undershot Wheels 360 

150. Vertical Impulse Wheels 363 

151. Horizontal Impulse Wheels ....... 366 

152. Downward-flow Wheels ....... . 369 

153. Special Forms 371 

Chapter XIV. Turbines 373 

Art. 154. The Reaction W T heel ........ 373 

155. Classification of Turbines ....... 376 

156. Reaction Turbines ......... 378 

157. Flow through Reaction Turbines ...... 3S2 

15S. Theory of Reaction Turbines . 385 

159. Design of Reaction Turbines ....... 389 

160. Guides and Vanes ......... 392 

161. Downward-flow Turbines ....... 394 

162. Impulse Turbines 397 

163. Special Devices ......... 399 

164. The Niagara Turbines ........ 402 

Appendix . 410 

Answers to Problems ......... 410 

Description of Tables 413 

Four-place Logarithms ......... 414 

Squares of Numbers ......... 416 

Areas of Circles .......... 418 

Friction Heads in Pipes .. ..... . 420 

Index 421 



Vlll 



CONTENTS. 



TABLES. 



I. Weight of Distilled Water . 
II. Atmospheric Pressure 

III. Heads and Pressures . 

IV. Theoretic Velocities 
V. Velocity Heads . 

VI. Coefficients for Circular Vertical Orifices 
VII. Coefficients for Square Vertical Orifices 
VIII. Coefficients for Rectangular Orifices, I foot 
IX. Coefficients for Submerged Orifices 
X. Coefficients for Contracted Weirs 
XI. Coefficients for Suppressed Weirs 
XII. Submerged Weirs 

XIII. Corrections for Wide Crests 

XIV. Coefficients for Conical Tubes 
XV. Vertical Heights of Jets from Nozzles 

XVI. Friction Factors for Pipes . 
XVII. Coefficients for Circular Conduits 
XVIII. Cross-sections in Circular Conduits 
XIX. Coefficients for Rectangular Conduits 
XX. Coefficients for Sewers 
XXI. Coefficients for Channels in Earth 
XXII. Values of the Backwater Function 
XXIII. Test of a 6-inch Eureka Turbine 
XXIV. Results of Test of a 6-inch Turbine 
XXV. Test of an 8o-inch Boyden Turbine 
XXVI. Four-place Logarithms 
XXVII. Squares of Numbers . 
XXVIII. Areas of Circles .... 
XXIX. Friction Heads for ioo Feet of Pipe 



wide 



PAGE 

6 

S 

17 
40 
40 

79 
81 

83 

87 

no 

113 
117 
120 
131 
135 
16S 
218 
222 
227 
233 

239 
2S0 
296 

297 
300 

414 
416 
41S 
420 



Evolvi varia problemata. In scientiis enim ediscendis prosunt exempla 
magis quam praecepta. Qua de causa in his fusius expatiatus sum. 

Newton. 



HYDRAULICS. 



CHAPTER I. 
INTRODUCTION. 

Article i. Units of Measure. 

The unit of linear measure universally adopted in English 
and American hydraulic literature is the foot, which is defined 
as one-third of the standard yard. For some minor purposes, 
such as the designation of the diameters of orifices and pipes, 
the inch is employed, but inches should always be reduced to 
feet for use in hydraulic formulas. The unit of superficial 
measure is usually the square foot, except for the expression of 
the intensity of pressures, when the square inch is more com- 
monly employed. 

The units of volume employed in measuring water are the 
cubic foot and the gallon. In Great Britain the Imperial gal- 
lon is used, and in this country the old English gallon, the 
former being 20 per cent larger than the latter. The following 
are the relations between the cubic foot and the two gallons : 

I cubic foot =6.232 Imp. gallons = 7.481 U. S. gallons; 
I Imp. gallon = 0.1605 cubic feet = 1.200 U. S. gallons; 
I U.S. gallon = 0.1337 cubic feet =0.8331 Imp. gallons. 

In this book the word gallon will always mean the United 
States gallon of 231 cubic inches, unless otherwise stated. 



2 INTRODUCTION. [Chap. I. 

The unit of weight is the avoirdupois pound, which is also 
the unit for measuring pressures. The intensity of pressure 
will be measured in pounds per square foot or in pounds per 
square inch, as may be most convenient, and sometimes in 
atmospheres (Art. 4). Gauges for recording the pressure of 
water are usually graduated so as to read pounds per square 
inch. 

The unit of time used in all hydraulic formulas is the second, 
although in numerical problems the time is often stated in 
minutes, hours, or days. Velocity is defined as the space passed 
over by a body in one second under the condition of uniform 
motion, so that velocities are to be always expressed in feet per 
second, or are to be reduced to these units if stated in miles per 
hour or otherwise. 

The unit of work, or energy, is the foot-pound ; that is, one 
pound lifted through a vertical distance of one foot. Energy 
is potential work, or the work which can be done ; for example, 
a moving stream of water has the ability to do a certain amount 
of work by virtue of its weight and velocity, and this is called 
energy, while the word work is more generally used for that 
actually done by a motor which is moved by the water. Power 
is work, or energy, done or existing in a specified time, and the 
unit for its measure is the horse-power, which is 550 foot-pounds 
per second, or 33 000 foot-pounds per minute. 

In French and German literature the metric system is em- 
ployed ; the meter and centimeter being the units of length, and 
their squares the units of superficial measure. The units of 
capacity are the cubic meter and the liter, that of weight the 
kilogram, and that of time the second. The unit of work is the 
kilogram-meter, and one horse-power is 75 kilogram-meters, 
which is about 1.5 per cent less than that as defined above. 
Students should be prepared to rapidly transform metric into 



Art. 2.] PHYSICAL PROPERTIES OF WATER. 3 

American measures, for which purpose a table of equivalents 
giving logarithms will be found most convenient.* 

The motion of water in river channels, and its flow through 
orifices and pipes, is produced by the force of gravity. This 
force is proportional to the acceleration of the velocity of a 
body falling freely in a vacuum ; that is, to the increase in 
velocity in one second. The acceleration is measured in feet 
per second per second, so that its value represents the number 
of feet per second which have been gained in one second by a 
falling body. 

Problem I. How many pounds per square inch are 
equivalent to a pressure of 70 kilograms per square centimeter? 

Article 2. Physical Properties of Water. 

At ordinary temperatures pure water is a colorless liquid 
which possesses perfect fluidity; that is, its particles have the 
capacity of moving over 

each other, so that the \ == Hi. ,^\ 

slightest disturbance of ^\ \ ^^^ / \ 

equilibrium causes a flow. \ ^A Vj L 

It is a consequence of this Ac 

property that the surface FlG - x - 

of still water is always level ; also, if several vessels or tubes be 
connected, as in Fig. 1, and water be poured into one of them, 
it rises in the others until, when equilibrium ensues, the free 
surfaces are in the same level plane. 

The free surface of water is in a different molecular condi- 
tion from the other portions, its particles being drawn together 
by stronger attractive forces, so as to form what may be called 
the " skin of the water," upon which insects walk. The skin is 
not immediately pierced by a sharp point which moves slowly 

* See Landreth's Metrical Tables for Engineers (Philadelphia, 1883). 



4 INTRODUCTION. [Chap. I. 

upward toward it, but a slight elevation occurs, and this prop- 
erty enables precise determinations of the level of still water 
to be made by means of the hook gauge (Art. 50). 

At about 32 degrees Fahrenheit a great alteration in the 
molecular constitution of water occurs, and ice is formed. If a 
quantity of water be kept in a perfectly quiet condition, it is 
found that its temperature can be reduced to 20 , or even to 
1 5 , Fahrenheit, before congelation takes place, but at the 
moment when this occurs the temperature rises to 32 . The 
freezing-point is hence not constant, but the melting-point of 
ice is always at the same temperature of 32 Fahrenheit or o° 
Centigrade. 

' Ice being lighter than water, forms as a rule upon its sur- 
face ; but when water is in rapid motion a variety called anchor 
ice may occur. In this case the ice is formed at the surface in 
the shape of small needles, which are quickly carried to the 
lower strata by the agitation due to the motion ; there the 
needles adhere to the bed of the stream, sometimes accumulat- 
ing to an extent sufficient to raise the water level several feet.* 
Anchor ice frequently causes obstructions in conduits and 
orifices which lead water to motors. 

Water is a solvent of high efficiency, and is therefore never 
found pure in nature. Descending in the form of rain it ab- 
sorbs dust and gaseous impurities from the atmosphere ; flow- 
ing over the surface of the earth it absorbs organic and mineral 
substances. These affect its weight only slightly as long as it 
remains fresh, but when it has reached the sea and become salt 
its weight is increased more than two per cent. The flow of 
water through orifices and pipes is only in a very slight degree 
affected by the impurities held in solution. 

* Francis in Transactions American Society Civil Engineers, 1S81, vol. x. 
p. 192. 



Art. 3.] THE WEIGHT OF WATER. 5 

The capacity of water for heat, the latent heat evolved when 
it freezes, and that absorbed when it is transformed into steam, 
need not be considered for the purposes of hydraulic investiga- 
tions. Other physical properties, such as its variation in volume 
with the temperature, its compressibility, and its capacity for 
transmitting pressures, are discussed in detail in the following 
pages. The laws which govern its pressure, flow, and energy 
under various circumstances belong to the science of Hydraulics, 
and form the subject-matter of this volume. 

Prob. 2. What horse-power is required to lift 16000 
pounds of water per minute through a vertical height of 21 
feet? Ans. 10.2. 

Article 3. The Weight of W t ater. 

The weight of water per unit of volume depends upon the 
temperature and upon its degree of purity. The following 
approximate values are, however, those generally employed 
except when great precision is required : 

1 cubic foot weighs 62.5 pounds; 
1 U. S. gallon weighs 8.355 pounds. 

These values will be used in this book, unless otherwise stated, 
in the solution of the examples and problems. 

The weight per unit of volume of pure distilled water is the 
greatest at the temperature of its maximum density, 39 . 3 
Fahrenheit, and least at the boiling-point. For ordinary com- 
putations the variation in weight due to temperature is not 
considered, but in tests of the efficiency of hydraulic motors 
and of pumps it should be regarded. The following table is 
hence given, which contains the weights of one cubic foot of 
pure water at different temperatures as deduced by Smith 
from the experiments of ROSSETTL* 

* Hamilton Smith, Jr., Hydraulics : The Flow of Water through Orifices, 
«over Weirs, and through open Conduits and Pipes (London and New York, 
1886), p. 14. 



INTRODUCTION. 



[Chap. 



TABLE I. WEIGHT OF DISTILLED WATER. 



Temperature 


Pounds per 


Temperature 


Pounds per 


Temperature 


Pounds per 


(Fahrenheit). 


Cubic Foot. 


(Fahrenheit). 


Cubic Foot. 


(Fahrenheit). 


Cubic Foot. 


32° 


62.42 


95 


62.06 


160 


61.OI 


35 


62.42 


IOO 


62.OO 


165 


60.90 


39-3 


62.424 


I05 


61.93 


170 


60.80 


45 


62.42 


110 


61.86 


175 


60.69 


SO 


62.41 


115 


61.79 


180 


60.59 


55 


62.39 


I20 


61.72 


185 


60.48 


60 


62.37 


125 


61.64 


190 


60.36 


65 


62.34 


130 


61.55 


195 


60.25 


7o 


62.30 


135 


61.47 


200 


60.14 


75 


62.26 


140 


61.39 


205 


60.02 


80 


62.22 


145 


6l .30 


2IO 


59-89 


85 


62.17 


I50 


61.20 


212 


59-84 


90 


62.12 


155 


6l. II 







Waters of rivers, springs, and lakes hold in suspension and 
solution inorganic matters which cause the weight per unit of 
volume to be slightly greater than for pure water. River 
waters are usually between 62.3 and 62.5 pounds per cubic foot, 
depending upon the amount of impurities and on the tempera- 
ture, while the water of some mineral springs has been found 
to be as high as 62.7. It appears that, in the absence of specific 
information regarding a particular water, the weight 62.5 pounds 
per cubic foot is a fair approximate value to use. It also has 
the advantage of being a convenient number in computations, for 
62.5 pounds is 1000 ounces, or ^-JiP" * s tne equivalent of 62.5. 

In the metric system the weight of a cubic meter of pure 
water at a temperature near that of maximum density is taken 
as 1000 kilograms, which is the average unit-weight used in 
hydraulic computations. This corresponds to 62.426 pounds 
per cubic foot. 

Brackish and salt waters are always much heavier than fresh 
water. For the Gulf of Mexico the weight per cubic foot is 
about 63.9, for the oceans about 64.1, while for the Dead Sea 
there is stated the value 73 pounds per cubic foot. The weight 
of ice per cubic foot varies from 57.2 to 57.5 pounds. 



Art. 4.] ATMOSPHERIC PRESSURE. 7 

Prob. 3. How many pounds of water in a cylindrical box 
2 feet in diameter and 2 feet deep ? How many gallons ? How 
many kilograms? How many liters? 

Prob. 4. In a certain problem regarding the horse-power 
required to lift water, the computations were made with the 
mean value 62.5 .pounds per cubic foot. Supposing that the 
actual weight per cubic foot was 62.35 pounds, show that the 
error thus introduced was less than one-fourth of one per cent. 



Article 4. Atmospheric Pressure. 

The pressure of the atmosphere is measured by the readings 
of the barometer. This instrument is a tube entirely exhausted 
of air, which is inserted into a vessel containing a liquid. The 
pressure of the air on the surface of the liquid causes it to rise 
in the tube until it attains a height which exactly balances the 
pressure of the air. Or in other words, the weight of the baro- 
metric column is equal to the weight of a column of air of the 
same cross-section as that of the tube, both columns being 
measured upward from the surface of the liquid in the vessel. 
The liquid generally employed is mercury, and, owing to its 
great density, the height of the column required to balance the 
atmospheric pressure is only about 30 inches, whereas a water 
barometer would require a height of over 30 feet. 

The atmosphere exerts its pressure with varying intensity, 
as indicated by the readings of the mercury barometer. At 
and near the sea level the average reading is 30 inches, and as 
mercury weighs 0.49 pounds per cubic inch at common tem- 
peratures, the average atmospheric pressure is taken to be 
30 X 0.49 or 14.7 pounds per square inch. 

The pressure of one atmosphere is therefore defined to be 
a pressure of 14.7 pounds per square inch. Then a pressure of 
two atmospheres is 29.4 pounds per square inch. And con- 



INTRODUCTION. 



[Chap, b 



versely, a pressure of one pound per square inch may be expressed 
as a pressure of 0.068 atmospheres. 

The rise of -water in a vacuum is due merely to the pressure 
of the atmosphere, like that of the mercury in the common 
barometer. In a perfect vacuum, water will rise to a height of 
about 34 feet under the mean pressure of one atmosphere, for 
the specific gravity of mercury is 13.6 times that of pure water, 
and as 30 inches is 2.5 feet, 13.6 X 2.5 = 34.0 feet. A water 
barometer is impracticable for use in measuring atmospheric 
pressures, but it is convenient to know its approximate height 
corresponding to a given height of the mercury barometer. 
The following table gives in the first column heights of the 
mercury barometer, in the second the corresponding pressures 
per square inch, in the third the pressures in atmospheres, and 
in the fourth the heights of the water barometer. This fourth 
column is computed by multiplying the numbers in the first 
column by 1.133, which is one-twelfth of 13.6, the specific 
gravity of mercury. 

TABLE II. ATMOSPHERIC PRESSURE. 



Mercury 
Barometer. 


"Pressure. 
Pounds per 


Pressure. 
Atmospheres. 


Water 
Barometer. 


Elevations. 
Feet. 


Boiling--point 
of Water 


Inches. 


Square Inch. 


Feet. 


(Fahrenheit). 


31 


15-2 


I.03 


35-1 


— S95 


2i3 3 -9 


30 


14-7 


I. 


34-0 


O 


212 .2 


29 


14.2 


O.97 


32.9 


+ 925 


2IO .4 


28 


13-7 


0.93 


3i-7 


1SS0 


203 .7 


27 


[3-2 


0.90 


30.6 


2S70 


206 .9 


26 


12.7 


0.S6 


29-5 


39°° 


205 .0 


25 


12 . 2 


0.83 


28. 3 


4970 


203 .1 


24 


11. 7 


0.80 


27. 2 


60S 5 


201 . 1 


23 


11. 3 


0.76 


26. 1 


7240 


199 .0 


22 


10.8 


0.72 


24.9 


8455 


196 .9 


21 


10.3 


0.69 


23. 8 


9720 


194 .7 


20 


9.8 


0.67 


22.7 


1 1050 


192 .4 



This table also giv 



ves in the fifth column values adapted 
from the vertical scale of altitudes used in barometric work, 
which show approximate vertical heights corresponding to 



Art. 5-] COMPRESSIBILITY OF WATER. 9 

barometer readings, provided that the pressure at sea level is 
30 inches.* In the last column are given the approximate 
boiling-points of water corresponding to the readings of the 
mercury barometer. 

Prob. 5. What pressure in pounds per square inch exists at 
the base of a column of water 170 feet high? What pressure 
in atmospheres ? 

Article 5. Compressibility of Water. 

The popular opinion that water is incompressible is not 
justified by experiments, which show in fact that it is more 
compressible than iron or even timber within the elastic limit. 
These experiments indicate that the amount of compression is 
directly proportional to the applied pressure, and that water is 
perfectly elastic, recovering its original form on the removal of 
the pressure, The amount of linear compression caused by a 
pressure of one atmosphere is, according to the measures of 
GRASSI, from 0.000051 at 35 Fahrenheit to 0.000045 at 8o° 
Fahrenheit. 

Taking 0.00005 as a mean value of the linear compression 
per atmosphere, the coefficient of elasticity of water is 

- 294 000 pounds per square inch, 



0.00005 

which is only one-fifth of the coefficient of elasticity of timber, 
and less than one-eightieth that of wrought-iron.f 

A column of water hence increases in density from the 
surface downward. If its weight at the surface be 62.5 pounds 
per cubic foot, at a depth of 34 feet a cubic foot will weigh 

62.5 (1 -f 0.00005) = 62.503 pounds, 

* Plympton, The Aneroid Barometer (New York, 1878). 

f Merriman's Mechanics of Materials (New York, 1885), p. 9. 



10 INTRODUCTION. [Chap. L' 

and at a depth of 340 feet a cubic foot will weigh 

62.5 (1 + 0.0005) = 62.53 pounds. . 

The variation in weight, due to compressibility, is hence toa 
small to be regarded in hydrostatic computations. 

Prob. 6. If w be the weight of water per cubic foot at 
the surface, show that the weight at a depth of d feet is. 
w (1 + 0.0000015 d). 

Article 6. The Acceleration of Gravity. 

The symbol g is used in hydraulics to denote the accelera- 
tion of gravity ; that is, the increase in velocity per second for 
a body falling freely in a vacuum at the surface of the earth. 
At the end of t seconds from the beginning of the fall, the 
velocity of the body is 

V= & t. 

The space, /i, passed over in this time, is the product of the 
mean velocity, \V, and the number of seconds, t, or 

h = fcA 

The relation between the velocity and the space is found by 
eliminating / from these two equations, and is 

V = Vigh. 

Hence the velocity of a body which has fallen freely through 
any height varies as the square root of that height. This equa- 
tion may also be written in the form 

which shows that the height or space varies with the square of 
the velocity of the falling body. 



Art. 6.] THE ACCELERATION OF GRAVITY. II 

The quantity 32.2 feet per second per second is an approxi- 
mate value of g which is often used in hydraulic formulas. It 
is, however, well known that the force of gravity is not of con- 
stant intensity over the earth's surface, but is greater at the poles 
than at the equator, and also greater at the sea level than on 
high mountains. The following formula of PEIRCE, * which is 
partly theoretical and partly empirical, gives the value of g in 
feet for any latitude /, and any elevation e above the sea level, 
e being taken in feet : 

g — 32.0894 (i -J- 0.0052375 sin 2 /)(i — 0.000000095 7^) : 
and from this its value may be computed for any locality. 

The greatest value of^is at the sea level at the pole, for 
which 

/ = 90 , e = o, whence g = 32.258. 

The least value of^-is on high mountains at the equator; for 
this there may be taken 

/=o°, <?= 10 000 feet, whence g = 32.059. 

Again, for the United States the practical limiting values are : 

/=49°, e = o, whence g= 32.186; 

/= 25 , e = 10 000 feet, whence g = 32.089. 

These results indicate that 32.2 feet is too large for a mean 
value of the acceleration. 

In the numerical work of this book, the value of the accel- 
eration is taken to be, unless otherwise stated, 

g = 32.16 feet per second per second, 

* Smith's Hydraulics, p. 19, where may be found a table giving values of 



12 INTRODUCTIOX. [Chap. 1. 

from which the frequently occurring quantity Vig is found 
to be 

V2g — 8.02. 

If greater precision be required, which will rarely be the case, 
g can be computed from the formula for the particular latitude 
and elevation above sea level. 

Prob. 7. Compute the value of g for the latitude 40 36', 
and the elevation 400 feet. 

Prob. 8. What is the value of g if the unit of time be one 
minute? Ans. 1 15 776 feet per minute per minute. 



Article 7. Numerical Computations. 

The numerical work of computation should not be carried 
to a greater degree of refinement than the data of the problem 
warrant. For instance, in questions relating to pressures, the 
data are uncertain in the third significant figure, and hence 
more figures than three or four in the final result must be 
delusive. Thus, let it be required to compute the number of 
pounds of water in a box containing 307.37 cubic feet. Taking 
the mean value 62.5 pounds as the weight of one cubic foot, 
the multiplication gives the result 19210.625 pounds, but 
evidently the decimals here have no precision, since the last 
figure in 62.5 is not accurate, and is likely to be less than 5, de- 
pending upon the impurity of the water and its temperature. 
The proper answer to this problem is 19 200 pounds, or per- 
haps 19 210 pounds, and this is to be regarded as a probable 
average result rather than an exact definite quantity. 

The use of logarithms is to be recommended in hydraulic 
computations, as thereby both mental labor and time are saved. 
Four-figure tables are sufficient for all common problems, and 
their use is particularly advantageous in cases where the data 
are not precise, as thus the number of significant figures in 



Art. 7.] NUMERICAL COMPUTATIONS. 1 3 

results is kept at about three and statements implying great 
precision, when none really exists, are prevented. In some 
problems five-figure logarithms will be needed, but probably no 
hydraulic data are ever sufficiently exact to require the use of 
a seven-figure table. Six-figure logarithms should not be em- 
ployed if others can be obtained, as their arrangement is not 
generally convenient for interpolation. 

As this book is mainly intended for the use of students in 
technical schools, a word of advice directed especially to them 
may not be inappropriate. It will be necessary for students in 
order to gain a clear understanding of hydraulic science, or of 
any other engineering subject, to solve many numerical prob- 
lems, and in this a neat and systematic method should be cul- 
tivated. The practice of performing computations on any loose 
scraps of paper that may happen to be at hand should not be 
followed, but the work should be done in a special book pro- 
vided for that purpose, and be accompanied by such explanatory 
remarks as may seem necessary in order to render the solution 
clear. Such a note-book, written in ink, and containing the 
fully worked out solutions of the problems and examples given 
in these pages, will prove of great value to every student who 
makes it. 

Prob. 9. Compute the weight of a column of water 1.1286 
inches in diameter and 34.0 feet high at the temperature of 
62 Fahrenheit. 

Prob. 10. How many gallons of water are contained in a 
pipe 4 inches in diameter and 12 feet long? How many 
pounds? 



14 HYDROSTATICS. [Chap. II. 



CHAPTER II. 
HYDROSTATICS. 

Article 8. Transmission of Pressures. 

One of the most remarkable properties of water is its 
capacity of transmitting a pressure, applied at one point of the 
surface of a closed vessel, unchanged in intensity, in all direc- 
tions, so that the effect of the applied pressure is to cause an 
equal force per square inch upon all parts of the enclosing sur- 
face. This is a consequence of the perfect fluidity of the water, 
by which its particles move freely over each other and thus 
transmit the applied pressure. 

An experimental proof of this property is seen in the hydro- 
static press, where the force applied to the small piston is ex- 
erted through the fluid and produces an equal unit-pressure 
at every point on the large piston. The applied force is here 
multiplied to any required extent, but the work performed by 
the large piston cannot exceed that imparted to the fluid by 
the small one. Let a and A be the areas of the small and 
large pistons, and p the pressure in pounds per square unit ap- 
plied to a ; then the total pressure on the small piston is pa, 
and that on the large piston is pA. Let the distances through 
which the pistons move at one stroke of the smaller be d and 
D. Then the imparted work is pad, and the performed work, 
neglecting hurtful resistances, is pAD. Consequently ad= AD, 
and since a is small as compared with A, the distance D must 



Art. 9.] HEAD AND PRESSURE. 1 5 

be small compared with d. Here is found an illustration of the 
popular maxim that " What is gained in force is lost in velocity." 

The pressure existing at any point within a body of water 
is exerted in all directions with equal intensity. This im- 
portant property follows at once from that of the transmission 
of pressure, for this may be regarded as effected by the con- 
fined body of water acting as an elastic spring which presses 
outwards in all directions. Thus every particle of the water is 
in a state of stress, and reacts in all directions with equal in- 
tensity. And the same principle applies to a particle within a 
body of water whose surface is free, for the pressure which ex- 
ists at any point due to the weight above it produces a state of 
stress among all the fluid particles. 

Prob. 11. In a hydrostatic press a work of one-fourth a 
horse-power is applied to the small piston. The diameter of 
the large piston is 12 inches, and it moves half an inch per 
minute. Find the pressure per square inch in the fluid. 

Ans. 1750 pounds. 

Article 9. Head and Pressure. 

The free surface of water at rest is perpendicular to the di- 
rection of the force of gravity, and for bodies of water of small 
extent this surface may be regarded as a plane. Any depth 
below this plane is called "a head," or the head upon any 
point is its vertical depth below the level surface. Let h be 
the head and w the weight of a cubic unit of water ; then at 
the depth Ji one horizontal square unit bears a pressure equal 
to the weight of a column of water whose height is h, and 
whose cross-section is one square unit, or wk. But the pres- 
sure at this point is exerted in all directions with equal inten- 
sity. The unit-pressure/ at the depth h then is 

p = wh\ (1) 



16 HYDROSTATICS. [Chap. II. 

and conversely the depth, or head, for a unit-pressure p is 

h=~ (I 7 ) 

If h be taken in feet and p in pounds per square foot, these 
formulas are 

p — 62.5/1, 
h = 0.016/. 

Hence pressure and head are mutually convertible, and in fact 
one is often used as synonymous with the other, although really 
each is proportional to the other. Any pressure / can be re- 
garded as produced by a head Ji, which sometimes is called the 
"pressure head." 

In numerical work p is usually taken in pounds per square 
inch, while h is expressed in feet. Thus, the pressure in pounds 
per square foot is 62.5/1, and the pressure in pounds per square 
inch is y^- of this ; or, 

p = 0.434/1, 
h = 2.304/). 

Stated in words these rules are : 

1 foot head corresponds to 0.434 pounds per square inch ; 
I pound per square inch corresponds to 2.304 feet head. 

These values, be it remembered, depend upon the assump- 
tion that 62.5 pounds is the weight of a cubic foot of water, 
and hence are liable to variation in the third significant figure 
(Art. 3). The extent of these variations for fresh water may 
be judged by the following table, which gives multiples of the 
above values, and also the corresponding quantities when the 
cubic foot is taken as 62.3 pounds. 



Art. 9.] HEAD AND PRESSURE. 

TABLE III. HEADS AND PRESSURES. 



17 



Head 
in Feet. 


Pressure in Pounds 
per Square Inch. 


Pressure 

in Pounds 

per Square 

Inch. 


Head in Feet. 


iv = 62.5 


iv = 62.3 


w = 62.5 


w = 62.3 


I 

2 

3 
4 

5 
6 

7 
8 

9 
10 


0-434 

O.868 
1 .302 
I.736 
2. 170 
2.604 
3-038 

3-4/2 
3.906 

4-340 


0.433 

0.865 
1.298 

I-73I 
2. 163 

2.596 
3.028 
3.461 

3-894 
4.326 


I 
2 

3 
4 
5 
6 

7 

8 

9 
10 

1 


2.304 

4.608 
6.912 
9.216 
II.520 
13.824 
16.128 
18.432 
20.736 
23.040 


2. 311 
4.623 

6-934 
9.246 

H-557 
13.868 
16.180 
18.491 
20.803 
23.114 



The atmospheric pressure, whose average value is 14.7 pounds 
per square inch, is transmitted through water, and is to be added 
to the pressure due to the head whenever it is necessary to 
regard the absolute pressure. This is important in some in- 
vestigations on the pumping of water, and in a few other cases 
where a partial or complete vacuum is produced on one side of 
a body of water. For example, if the air be exhausted from a 
small globe, so that its tension is only 5 pounds per square 
inch, and it be submerged in water to a depth of 250 feet, the 
absolute pressure per square inch on the globe is 

p — 0.434 X 250+ 14.7 = 123.2 pounds, 

and the resultant effective pressure per square inch is 

p' = 123.2— 5.0 = 1 18.2 pounds. 

Unless otherwise stated, however, the atmospheric pressure 
need not be regarded, since under ordinary conditions it acts 
with equal intensity upon both sides of a submerged surface. 

Prob. 12. What unit pressure corresponds to 230 feet head? 
What head in meters produces a pressure of 10 kilograms per 
square centimeter? 



i8 



HYDROSTA TICS. 



[Chap. II. 



Prob. 13. The pressure in a water pipe in the basement is 
74I pounds, while in the fifth story it is only 48 pounds per 
square inch. Find the height of the fifth story above the 
basement. 




Fig. 2. 



Article 10. Normal Pressure. 

The total normal pressure on any submerged surface may be 
found by the following theorem : 

The normal pressure is equal to the product of the weight 
of a cubic unit of water, the area of the surface, and the 
head on its centre of gravity. 

To prove this let A be the area of the surface, and imagine 

it to be composed of elemen- 
tary areas, a x , a %1 a 31 etc.. 
each of which is so small that 
the unit-pressure over it may 
be taken as uniform ; let 
h x , // 2 , h % , etc., be the heads 
on these elementary areas, 
and let w denote the weight of a cubic unit of water. The 
unit-pressures at the depths h x , /i 2 , h % , etc., are wh x , wh % , wk a , 
etc. (Art. 9), and hence the normal pressures on the elementary 
areas a lf a 2 , a % , etc., are wajt x , zvaji, , wa % h % , etc. The total 
normal pressure P on the entire surface then is 

P = w(a 1 k 1 + aji, -f a 3 /i 3 + etc.). 

Now let h be the head on the centre of gravity of the surface ; 
then, from the definition of the centre of gravity, 

a J h l -\- aji. z -\- aji % -\- etc. = Ah. 

Therefore the normal pressure is 

P— wAk, (2)' 

which proves the theorem as stated. 



Art. io. J NORMAL PRESSURE. 1 9 

This rule applies to all surfaces, whether plane, curved, or 
warped, and however they be situated with reference to the 
water surface. Thus the total normal pressure upon the sur- 
face of a submerged cylinder remains the same whatever be its 
position, provided the depth of the centre of gravity of that 
surface be kept constant. It is best to take h in feet, A in 
square feet, and w as 62.5 ; then Pwill be in pounds. In case 
surfaces are given whose centres of gravity are difficult to de- 
termine, they should be divided into simpler surfaces, and then 
the total normal pressure is the sum of the normal pressures on 
the separate surfaces. 

The normal pressure on the base of a vessel filled with water 
is equal to the weight of a cylinder of water whose base is the 
base of the vessel, and whose height is the depth of water, and 
only in the case of a vertical cylinder does this become equal to 
the weight of the water. Thus the pressure on the base of a 
vessel depends upon the depth of water and not upon the 
shape of the vessel. Also in the case of a dam, the depth of 
the water and not the size of the pond determines the amount 
of pressure. 

The normal pressure on the interior surface of a sphere filled 
with water is greater than the weight of the water, for the 
weight acts only vertically, while the normal pressures are ex- 
erted in all directions. If d be the diameter of the sphere, for- 
mula (2) gives 

P = w nd" 1 • \d = ^W7td 3 f 
while the weight of water is 

W= w\ud* = \wnd\ 

Hence the interior normal pressure is three times the weight 
of the water. 

Prob. 14. A cone with altitude h and diameter of base d is 
filled with water. Find the normal pressure on the interior 



20 HYDROSTATICS. [Chap. II. 

surface (a) when it is held vertical with base downward ; (b) when 
held horizontal. 

Prob. 15. A board 2 feet wide at one end and 2 feet 6 
inches at the other is 8 feet long. What is the normal pres- 
sure upon each of its sides when placed vertically in water with 
the narrow end in the surface? 

Article ii. Pressure in a Given Direction. 

The pressure against a submerged plane surface in a given 
direction may be found by obtaining the normal pressure by 
Art. 10 and computing its component in the required direc- 
tion, or by means of the following theorem : 

The horizontal pressure on any plane surface is equal to 
the normal pressure on its vertical projection ; the 
vertical pressure is equal to the normal pressure on its 
horizontal projection ; and the pressure in any direction 
is equal to the normal pressure on a projection perpen- 
dicular to that direction. 

To prove this let A be the area of the given surface, repre- 
sented by AA in Fig. 3, and 
P the normal pressure upon 
it, or P — wAh. Now let it 
be required to find the pres- 
sure P' in a direction mak- 
ing an angle 6 with the 
normal to the given plane. 
Draw A' A' perpendicular to 
the direction of P, and let 

A' be the area of the projection of A upon it. The value of 

P then is 

P' = P cos 6 = wAh cos e. 

But A cos 6 is the value of A' by the construction. Hence 

P = wA'k, (3) 

and the theorem is thus demonstrated. 




ART. II.] 



PRESSURE IN A GIVEN DIRECTION. 



21 




Fig. 



This theorem does not in general apply to curved surfaces. 
But in cases where the head of 
water is so great that the pressure 
may be regarded as uniform it is 
also true for curved surfaces. For 
instance, consider a cylinder or 
sphere subjected on every ele- 
mentary area to the unit-pressure 
p due to the high head k, and let 
it be required to find the pressure 
in the direction shown by q x , q 2 , 
and q 3 in Fig. 4. The pressures A > A > A > etc -> on the ele- 
mentary areas a x , a 2 , a 3 , etc., are 

A = M > A = M > A = P a z , etc., 
and the components of these in the given direction are 

q. = pa x cos 1 , q 2 = pa 2 cos 2 , q % — pa 3 cos 9 Z , etc., 

whence the total pressure P' in the given direction is 

P' = p{a x cos ^ -f- a 2 cos ^ 2 -f- « 8 cos 6 5 -f- etc.). 

But the quantity in the parenthesis is the projection of the sur- 
face on a plane perpendicular to the given direction, or MN. 
Hence 

P' — p x area MN, 

which is the same rule as for plane surfaces. 

For the case of a water-pipe let p be the interior pressure per 
square inch, and d its diameter in inches. Then for a length 
of one inch the force tending to rupture the pipe longitudinally 
is pd. This is resisted by the unit stress 5 in the walls of the 
pipe acting over the area 2t, if t be the thickness. As these 
forces are equal, 

2St = pd, 



22 



HYDROSTATICS. 



[Chap. II. 



which is the fundamental equation for the discussion of the 
strength of water-pipes. 

Prob. 16. The back of a dam has a slope of \\ to i. Find 
the horizontal pressure per linear foot upon it, the water being 
13 feet deep. 

Prob. 17. What head of water will burst a pipe 24 inches 
in interior diameter and 0.75 inches thick, the tensile strength 
of the cast-iron being 20,000 pounds per square inch ? 



Article 12. Centre of Pressure on Rectangles. 

The centre of pressure on a surface submerged in water is 
the point of application of the resultant of all the normal pres- 
sures upon it. The simplest and probably the most important 
case is the following : 

If a rectangle be placed with one end in the water surface, 
the centre of pressure is distant from that end two- 
thirds of its length. 

This theorem will be proved by the help of the graphical 
illustration shown in Fig. 5. The rectangle, which in practice 

might be a board, is placed with 
its breadth perpendicular to the 
plane of the drawing, so that 
AB represents its edge. It is 
required to find the centre of 
pressure C. For any head h the 
unit-pressure is wh (Art. 9), and 
hence the unit-pressures on one 
side of AB may be graphically 
represented by arrows which form a triangle. Now if a force 
P equal to the total pressure is applied on the other side of 
the rectangle to balance these unit-pressures, it must be placed 
opposite to the centre of gravity of the triangle. Therefore 




Fig. 



Art. 12.] CENTRE OF PRESSURE ON RECTANGLES. 



23 




AC equals two-thirds of AB, and the rule is proved. The head 
on C is evidently also two-thirds of the head on B. 

Another case is that shown in Fig. 6, where the rectangle, 
whose length is B X B,, is wholly immersed, the head on B x being 
h l , and on i> 2 being /i 2 . Let 
AB 1 = b x , A C = j/, and AB, 
= b v Now the normal pres- 
sure P x on AB X is applied at 
the distance %b x from A, 
and the normal pressure P a 
on AB, is applied at the dis- 
tance \b, from A. The normal pressure P on B X B % is the dif- 
ference of P 1 and P, , or 

P = P,-P 1 ; 

and also, by taking moments about A as a centre, 

Now, by Art. 10, the values of P 2 and P x are, for a rectangle one 
unit in breadth, 

P 2 = wX&,X U,, P x = w Xb x X \K ; 



Fig. 6. 



hence 



-U 



w{bji % — bji x ) ; 



and inserting these in the equation of moments, the value of 
^is 

2 b?K - b?K 



y 



3 4A 



bX 



Now if 6 be the angle of inclination of the plane to the water 
surface, h 2 = b 2 sin 0, and h x = b x sin 0. Accordingly, the ex- 
pression becomes 



y = 7 



2 b: - b; 



ib? -by 



24 HYDROSTATICS. [Chap. II. 

Again, if h! be the head on the centre of pressure^ = h! cosec #, 
b. 2 = /i 2 cosec 6, and b l = h x cosec 6. These inserted in the 
last equation give 

2 h: - h? 



ti = 



zK-hy 



These formulas are very convenient for computation, as the 
squares and cubes may be taken from tables. 

If h x equals h 2 the above formula becomes indeterminate, 
which is due to the existence of the common factor /i 2 — k t in 
both numerator and denominator of the fraction ; dividing out 
this common factor, it becomes 

2 yfe. 1 -4- >Ufc, 4- A, a 



3 K + K ' 
from which, if /i 2 =/i x = h, there is found the result h! = h. 

If h, = o, or b 1 = o,y becomes f^ a and h! becomes f// 2 , 
which proves again the special rule given at the beginning of 
this article. 

Prob. 1 8. A rectangle 4 feet long is immersed in water with 
its ends parallel to the surface, the head on one end being 7 feet 
and that on the other 9 feet. Find the head on the centre of 
pressure, and also the value of P. 

Article 13. General Rule for Centre of Pressure. 

For any plane surface submerged in a liquid, the centre of 
pressure may be found by the following rule : 

Find the moment of inertia of the surface and its statical 
moment, both with reference to an axis siruated at the 
intersection of the plane of the surface with the water 
level. Divide the former by the latter, and the quotient 
is the perpendicular distance from that axis to the 
centre of pressure. 



Art. 13.] GENERAL RULE FOR CENTRE OF PRESSURE. 2$ 

The demonstration is analogous to that in the last article. 
Let, in Fig. 6, B X B^ be the trace of the plane surface, which 
itself is perpendicular to the plane of the drawing, and C be 
the centre of pressure, at a distance y from A where the plane 
of the surface intersects the water level. Let a x , a,, a 3 , etc., be 
elementary areas of the surface, and 1i x , /i 2 , Ji 3 , etc., the heads 
upon them, which produce the normal elementary pressures, 
<uuaji x , waji^ , wa 3 h 3 , etc. Let y x , y 2 , y 3 , etc., be the distances 
from A to these elementary areas. Then taking the point A 
as a centre of moments, the definition of centre of pressure 
gives the equation 

{waji x + waji 2 -f- waji 3 -\- etc.) y = 

wa 1 /i 1 y 1 -f- wa 2 /i 2 y 2 + waji % y % -\- etc. 

Now let 6 be the angle of inclination of the surface to the 
water level ; then k, = y 1 sin 6, // 2 = j/ 2 sin 6, h z = j/ 3 sin 6, etc. 
Hence, inserting these values, the expression for y is 

__ a 1 y* + a 9 y i * + * z y*+ etc. 
y ~ ' ^Ji + a % y % + a 3 y 3 + etc. ' 

The numerator of this fraction is the sum of the products 
obtained by multiplying each element of the surface by the 
square of its distance from the axis, which is called the moment 
of inertia of the surface. And the denominator is the sum of 
the products of each element of the surface by its distance from 
the axis, which is called the statical moment of the surface. 
Therefore 

moment of inertia I' 
•* ~ statical moment _ 5 ^ ' 

is the general rule for finding centres of pressure for plane 
surfaces. 

The statical moment of a surface is simply its area multi- 
plied by the distance of its centre of gravity from the given 



26 HYDRO ST A TICS. [Chap. II. 

axis, as is evident from the definition of centre of gravity. The 
moments of inertia of plane surfaces with reference to an axis 
through the centre of gravity are deduced in works on theo- 
retical mechanics; a few values are : 

bd 3 
For a rectangle of breadth b and depth d, I ' = ; 

bd 3 
For a triangle with base b and altitude d, I = —=■ ; 

nd* 
For a circle with diameter d, I = -r- . 

64 

To find from these the moment of inertia with reference to a 
parallel axis, the well-known formula F = I-\-Ak 2 is to be used, 
where A is the area of the surface and k the distance from the 
given axis to the centre of gravity of the surface, and F the 
moment of inertia required. 

For example, let it be required to find the centre of pres- 
sure of a circle which is submerged with one edge in the water 
surface. The area of the circle is ^yrd 2 , and its statical moment 
with reference to the upper edge is \rtd 2 X id. Then from (4), 



y = 



hence the centre of pressure of a circle with one edge in the 
water surface is at \d below the centre. Again, for a triangle 
submerged with its vertex in the water surface, 

bjF bd 4*P 

36 2 *9 3 ■, 

v = — ^— = -d. 

b A ?£ 4 

2 ■ 3~ 



nd' ltd* d 2 

"64 ' 4 ' "4 


-y> 


nd' d 
~4~*2 



Art. 14.] PRESSURES ON OPPOSITE SIDES OF A PLANE. 



2/ 



Prob. 19. Find the centre of pressure of the triangle in 
Fig. 9 when it is inverted so that its base is in the surface. 

Prob. 20. Find the centre of pressure of a circle when verti- 
cally submerged in water so that the head on its centre is equal 
to two diameters of the circle. Ans. 2.03^. 




Article 14. Pressures on Opposite Sides of a Plane. 

In the case of an immersed plane the water presses equally 
upon both sides so that no disturbance of the equilibrium re- 
sults from the pressure. But in case 
the water is at different levels on op- 
posite sides of the surface the opposing 
pressures are unequal. For example, 
the cross-section of a self-acting tide- 
gate, built to drain a salt marsh, is 
shown in Fig. 7. On the ocean side 
there is a head of k x above the sill, 
which gives for every linear foot of 
the gate the pressure 

which is applied at the distance \h x above the sill. On the 
other side the head on the sill is /z 2 , which gives the pressure 

whose centre of pressure is at \h % above the sill. The result- 
ant pressure P is 

and if z be the distance of the point of application of P above 
the sill, the equation of moments is 

(/>, - P,)s =P t X ¥h -P.Y. **, , 

from which z can be computed. 



Fig. 7. 



2$ 



HYDROSTATICS. 



[Chap. II. 



The action of the gate in resisting the water pressure is like 
that of a beam under its load, the two points of support being 
at the sill and the hinge. If h be height of the gate, the reac- 
tion at the hinge is, 

and this has its greatest value when h x becomes equal to k, 
and h. 2 is zero. In the case of the vertical gate of a canal lock, 
which swings horizontally like a door, a similar problem arises 

and a similar conclusion results. 

Prob. 21. If the head on one side of a tide-gate is J feet 
and on the other 4 feet, find the resultant pressure and its 
point of application above the sill. 

Ans. 103 1 pounds per linear foot, at 2.82 feet above sill. 



Article 15. Masonry Dams. 

The preceding articles show that the pressure on the back 
of a masonry dam is normal to that surface at every point. If 

the back be a plane surface the 
resultant pressure is normal to 
; - — ~ the plane, and its point of applica- 
tion is at two-thirds of the length 
from the water level. Thus in 
Fig. 8, A C is two-thirds of AB. If 
h be the head of water above the 
base of the dam. and t> be the 
angle of inclination of the plane of 
the back to the vertical, the normal pressure per linear foot of 
the dam is, from Art. 10, 

P=wX h sec & X \h = iwk* sec 0, 

which shows that the total pressure against the dam varies as 
the square of its height. The horizontal component of this 




Fig. 8. 



Art. 15.] MASONRY DAMS. 29 

pressure is \wff, which is the same as the normal pressure 
against a wall whose back is vertical. 

It is not the place here to enter into the discussion of the 
subject of the design of masonry dams, but the two ways in 
which they are liable to fail maybe noted. The first is that of 
sliding along a horizontal joint, as BD ; here the horizontal 
component of the thrust overcomes the resisting force of friction 
acting along the joint. If W be the weight of masonry above 
the joint, and f the coefficient of friction, the resisting friction 
is fW, and the dam will slide if the horizontal component of 
the pressure is equal to or greater than this. The condition 
for failure by sliding then is 

The second method of failure is that of rotating around the 
toe D\ this occurs when the moment of P equals the moment 
of PFwith reference to that point; or if / and m be the lever- 
arms dropped from D upon the directions of P and W y the 
condition for failure by rotation is 

PI = Wm. 

In practice the joints are so built as to give full security 
against sliding, so that the usual method of failure is by 
rotation. 

As an example of the application of these principles con- 
sider a rectangular vertical masonry dam which weighs 140 
pounds per cubic foot, and which is 4 feet wide. First, let it 
be required to find the height for which it would fail by slid- 
ing, the coefficient of friction being 0.75. The horizontal 
water pressure is \ X 62.5 X h*, and the resisting friction is 
0.75 X 140 X 4 X h. Placing these equal, there is found h = 
13.4 feet. Secondly, to find the height for which failure will 
occur by rotation, the equation of moments is stated with ref- 



30 HYDROSTATICS. [Chap. II. 

erence to the front lower edge, the lever-arm of the pressure 
being \h, and that of the wall 2 feet. Hence 

i X 62.5 X 1? X \h =140X4X^X2, 
from which there is found h = 10.4 feet. 

Prob. 22. A dam whose cross-section is a triangle has a 
vertical back, is 3 feet wide at the base, and 15 feet high. 
Find the height to which the water may rise behind it in order 
to cause failure (a) by sliding, and (&) by rotation, using 0.75 
for the coefficient of friction and 140 pounds per cubic foot for 
the weight of the masonry. 

Article 16. Loss of Weight in Water. 

It is a familiar fact that bodies submerged in water lose 
part of their weight a man can carry under water a large 
stone which would be difficult to lift in air ; timber when sub- 
merged has a negative weight or tends to rise to the surface. 
The following is the law of loss : * 

The weight of a body submerged in water is less than its 
weight in air by the weight of a volume of water equal 
to that of the body. 

To demonstrate this, consider that the submerged body 
is acted upon by the water pressure in all directions, and 
that the horizontal components of these pressures must bal- 
ance. Any vertical elementary prism is subjected to an up- 
ward pressure upon its base which is greater than the down- 
ward pressure upon its top, since these pressures are due to 
the heads. Let k t be the head on the top of the elementary 
prism and 1i n _ that on its base, and a the cross-section of the 
prism ; then the downward pressure is wah A and the upward 
pressure is wak 2 . . The difference of these, wa(k t — /i } ) is the re- 
sultant upward water pressure, and this is equal to the weight 
of a column of water whose cross-section is a and whose height 



Discovered by Archimedes, about 250 B.C. 



Art. 17. J 



DEPTH OF FLOTATION. 



31 



is that of the elementary prism. Extending this to all the 
■elementary prisms which make up the body, it is seen that the 
upward water pressure diminishes its weight by the weight of 
a volume of water equal to that of the body. 

It is important to regard this loss of weight in constructions 
under water. If, for example, a dam of loose stones allows the 
water to percolate through it, its weight per cubic foot is less 
than its weight in air, so that it can be more easily moved by 
horizontal forces. As stone weighs about 150 pounds per cubic 
foot in air, its weight in water is only about 150—62 — 88 
pounds. 

Prob. 23. A bar of iron one square inch in cross-section 
and one yard long weighs 10 pounds in air. What is its weight 
in water ? 

Article 17. Depth of Flotation. 

When a body floats upon water it is sustained by an upward 
pressure of the water equal to its own weight, and this pressure 
is the same as the weight of the volume of water displaced by 
the body. Let W be the weight of the floating body in air, 
and Wbe the weight of the displaced water; then 

W =W. (5) 

Now let z be the depth of flotation of the body; then to find 
its value for any particular case W is to be expressed in terms 
of the linear dimensions of the body, and W \n terms of the 
depth of flotation z. 

For example, a cone which weighs w' pounds per cubic foot 

floats with its base downward as 

represented in Fig. 9, its altitude 

being d and the radius of its base b. 

The weight of the floating cone is 

W = w' • ntf • id, 

Fig. 9 . and the weight of the displaced 

water is that of a frustum of the altitude z, or 




32 



HYDROSTATICS. 



[Chap. l\. 



Equating these values and solving for z gives the result 

z — d — d\ i — — j , 

which is the depth of flotation. If w r == w 9 the cone has the 
same density as water, and z = d\ if w' = o, the cone has no 

weight, and z = o. 

To find the depth of flotation 
for a cylinder lying horizontally, 
,A ^-^T"~^-^3/ let w' be its weight per cubic 

foot, and r the radius of its cross- 
section. The depth of flotation 
is DE (Fig. 10), or if 6 be the 
angle ACE, 

z = r{i — cos 6). 

The weight of the cylinder for one unit of length is 

W = w' . nr\ 

and that of the displaced water is 

IV = w (r 2 arc 6 — f 2 sin cos ff). 

Equating the values of W and IV , and substituting for 
sin cos its equivalent i sin 20, there results 




2 arc — sin 2d = 2n 



w 



From this equation 6 is to be found by trial for any particular 



Art. i8.3 STABILITY OF FLOTATION. 33 

case, and then z is known. For example, if w' — 26.5 pounds 
per cubic foot, and r — 12 inches, 

2 arc — sin 20 — 2.664 = o. 

To solve this equation, assume values for 0, until finally one is 
found that satisfies it ; thus : 

For = 83 , 2.897 — 0.242 — 2.664 = — 0.009 J 
For = 834, 2.906 — 0.234 — 2.664 = + 0.008. 

Therefore lies between 83 and 83 15' , and is probably about 
83 8\ Hence the depth of flotation is z = 12(1 —0.120) = 
I0.6 inches. 

Prob. 24. Show that the depth of flotation for a sphere 
whose radius is r is the real root of the cubic equation 

z* — V'z + 4r 3 — = O. 

Prob. 25. A rectangular wooden box 4.5 feet long, 3 feet 
wide, and 2.5 feet deep, inside dimensions, is made of timber 
\\ inches thick, which weighs 3 pounds per foot board measure. 
How much water will it draw when a weight of 200 pounds is 
placed in it and the cover nailed on ? Ans. 0.46 feet. 



Article 18. Stability of Flotation. 

The equilibrium of a floating body is stable when it returns 
to its primitive position after having been slightly moved there- 
from by extraneous forces, it is indifferent when it floats in any 
position, and it is unstable when the slightest force causes it to 
leave its position of flotation. For instance, a short cylinder 
with its axis vertical floats in stable equilibrium, but a long 
cylinder in this position is unstable, and a slight force causes it 
to fall over and float with its axis horizontal in indifferent 
equilibrium. 



34 



HYDRO STA TICS. 



[Chap, IL. 




Fig. ii. 



The stability depends in any case upon the relative posi- 
tion of the centre of gravity of the body and its centre of 
buoyancy, the latter being the centre of gravity of the dis- 
placed water. Thus in Fig. u let G be the centre of gravity 

of the body and C that of the 
centre of buoyancy when in an 
upright position. Now if an ex- 
traneous force causes the body 
to tip into the position shown, 
the centre of gravity remains 
at G, but the centre of buoy- 
ancy moves to D. In this new 
position of the body it is acted 
upon by the forces W and W, whose lines of direction pass 
through G and D. W is the weight of the body and W the 
weight of the displaced water ; and as these are equal, they 
form a couple which tends either to restore the body to the 
upright position or to cause it to deviate farther from that 
position. Let the vertical through D be produced to meet the 
centre line CG in M. If M is above G the equilibrium is stable, 
as the forces Wand W tend to restore it to its primitive posi- 
tion ; if M coincides with G the equilibrium is indifferent ; and 
if M be below G the equilibrium is unstable. 

The point M is often called the 'metacentre,' and the 
theorem may be stated that the equilibrium is stable, indifferent, 
or unstable according as the metacentre is above, coincident 
with, or below the centre of gravity of the body. The measure 
of the stability of a floating body is the moment of the couple 
formed by the forces IF and W. But the line G 31 is propor- 
tional to the lever arm of the couple, and hence the quantity 
Wx GM may be taken as a measure of the stability. The 
stability, therefore, increases with the weight of the body, and 
with the distance of the metacentre above the centre of gravity. 



Art. 18 ] STABILITY OF FLOTATION. 35 

To ensure a high degree of stability the centre of gravity should 
be as low as possible. 

The only important applications of these principles are in 
connection with the subject of naval architecture, and in general 
the resulting investigations are of a complex character, which 
can only be solved by approximate tentative methods. Reed's 
Treatise on the Stability of Ships (London, 1885) is a large 
volume entirely devoted to this topic. 

Prob. 26. If 6 be the angle of inclination to the vertical, c 
the distance between the metacentre and centre of gravity, 
show that the stability of flotation can be measured by the 
quantity Wc sin 0, 



36 THEORETICAL HYDRAULICS. [Chap. III. 



CHAPTER III. 

THEORETICAL HYDRAULICS. 

article 19. Velocity and Discharge. 

If a vessel or pipe be constantly full of water, all the parti- 
cles of which move with the same uniform velocity v, and if a 
be the area of its cross-section, the quantity of water which 
passes any section per second is equal to the volume of a prism 
whose base is a and whose length is v, or 



t> 



q— av (6) 

If, now, the vessel varies in cross-section, one area being a y 
another a 1 , and a third a 2 , the same quantity of water passes 
each section per second if the vessel be kept constantly full ; 
hence if v, v 1 , and v 2 be the respective velocities, 



av 



The velocities of flow in different sections of a pipe or vessel 
which is maintained constantly full hence vary inversely as 
the areas of the cross-sections. 

In case the particles or filaments move with different veloci- 
ties in different parts of the section, the quantity may be still 



Art. 20.] VELOCITY OF FLOW FROM ORIFICES. 37 

expressed by q = av y provided that v signifies the mean velocity 
of the flow ; or 

•=i ^ 

may be regarded as a definition of the term mean velocity. 

The word discharge will be used to denote the quantity of 
water flowing per second from a pipe or orifice, and the letter 
Q will designate the theoretic discharge, that is, the discharge 
as computed by t]je methods of this chapter, where resistances 
or losses due to friction, contraction, and other causes are not 
considered. The letter Fwill designate the theoretic velocity, 
so that if a be the area of an orifice, or the cross-section of a jet, 

is the formula for the theoretic discharge. In the case of flow 
from a simple orifice the area a is found by the measurement 
of its dimensions, so that the problem of finding Q is reduced 
to that of determining V. 

Prob. 27. A pipe constantly filled with water discharges 
0.43 cubic feet per second. Compute the mean velocity of flow 
if the pipe is 3 inches in diameter ; also if it is 6 inches in 
diameter. 

Article 20. Velocity of Flow from Orifices. 

If an orifice be opened, either in the base or side of a vessel 
containing water, it flows out with a velocity which is greater 
for high heads of water than for low heads. The theoretic 
velocity of flow is given by the theorem discovered by TORRI- 
celli* 

The theoretic velocity of flow at the orifice is the same as 
that acquired by a body falling freely in a vacuum 
through a height equal to the head of water on the 
orifice. 

*Del moto dei gravi (Firenz, 1644). 



38 



THE ORE TICAL H YDRA ULICS. 



[Chap. III. 




The proof of this rests partly on observation. Thus if a vessel be 

arranged, as in Fig. 12, so that 
a jet of water from an orifice is 
directed vertically upward, it is 
known that it never attains to 
the height of the level of the 
water in the vessel, although 
under favorable conditions it 
Fig. 12. nearly reaches that level. It 

may hence be inferred that the jet would actually rise to that 
height were it not for the resistance of the air and the friction 
of the edges of the orifice. Xow, since the velocity of impulse 
required to raise a body vertically to a certain height is the 
same as that acquired by it in falling from rest through that 
height, it is regarded as established that the velocity at the 
orifice is as stated in the theorem. 

The following proof rests on the law of conservation of 
energy. Let, as in the second diagram of Fig. 12, the water 
surface in a vessel be at A at the beginning of a second and at 
A 1 at the end of the second. Let /The the weight of water 
between the planes A and A 1} which is evidently the same as 
that which flows from the orifice during the second. Let U\ be 
the weight of water between the planes A 1 and B, and k t the 
height of its centre of gravity above the orifice. Let Ji be the 
height of A above the orifice, and 6h the distance between A 
and A l . Then at the beginning of the second the water in the 
vessel has the energy WJt^ + W(h — \Sh). If Tbe the velocity 
of flow through the orifice, the same water at the end of the 

second has the energy WJi x -f W— . By the law of conserva- 

tion these are equal, if no energy has been dissipated in friction 

or in other ways ; thus, 

V 2 
h - \8h = — . 



Art. 20.] VELOCITY OF FLOW FROM ORIFICES. 39 

Now if $k be small compared with h, which will be the case 
when A is large compared with the area of the orifice, this gives 

1 V= V'^k, 

which is the same as for a body falling freely through 
the height h (Art. 6). 

The theoretic velocity of flow from any orifice, whether its 
plane be horizontal^ vertical, or inclined, is thus given by 

V=V5£k, (7) 

provided the orifice be small compared with the section of the 
reservoir. The theoretic height to which the jet will rise is 

k = — (7)' 

The first of these formulas states the velocity due to a given 
head, and the second states the head which would generate a 
given velocity. The term "velocity head " will be generally 

used to designate the expression — ■, meaning thereby that its 

2g 

value is the head which can produce the velocity V, 

Using for g the mean value 32.16 feet per second per 
second, these formulas become 

V— 8.02 VJi, h = 0.01555 V\ 

from which the following tables have been computed. These 
are mainly intended to impress upon the student the fact that 
small heads produce rapid velocities, but they may also prove 
serviceable for use in approximate computations. The last 
columns of the tables give multiples of the numbers 8.02 
and 0.01555. 



40 



THE ORE TIC A L H YD RA ULICS. [Chap. III. 

TABLE IV. THEORETIC VELOCITIES. 



Head. 
Feet. 


Velocity. 
Feet per 
Second. 


Head. 
Feet. 


Velocity. 
Feet per 
Second. 


Multiples of 
8.01997. 


O.OOI 
.002 
.003 
.004 
.005 
.006 
.007 
.008 
.009 


O.254 
.358 
•439 
.507 
.567 
.621 
.671 
.717 
.761 


I 
2 
3 
4 
5 
6 

7 
8 

9 


8.02 

H-33 

13.89 
16.04 

17-93 
19.64 
21.22 
22.68 
24.06 


2 

3 
4 

5 
6 

7 
8 

9 


8.02 
16.04 
24.06 
32.08 
40.10 
48. 12 
56.14 
64.16 
72.18 



TABLE V. VELOCITY HEADS. 



Velocity. 
Feet per 
Second. 


Head. 
Feet. 


Velocity. 
Feet per 
Second. 


Head. 
Feet. 


Multiples of 
0.015547. 


I 
2 
3 

4 

5 
6 

7 
8 

9 


O.OI6 
.062 
.140 
.249 
.389 
.560 
.762 

•995 
1.260 


IO 
20 
30 
40 
50 
60 
70 
80 

90 


1-55 
6.22 

13-99 

24. 8S 
38.86 

55-97 
76.19 

99-51 
125-95 


I 
2 
3 

4 

5 
6 

7 

8 

9 


0.01555 
.03109 
.04664 
.06219 

•07774 
•09328 
.10883 

.1243S 
.13992 



Prob. 28. Find the theoretic velocity of flow from an orifice 
under a head of 6 inches. Find the velocity-head of a stream 
0.1 feet in diameter which discharges 2.5 cubic feet per minute- 
Ans. V— 5.67 feet per second, H — 0.44 feet. 

Article 21. Horizontal Orifices. 

Let a be the area of an orifice whose plane is horizontal, h 
the head of water upon it, and Q the quantity of water dis- 
charged per second. The theoretic discharge is, from the prin- 
ciples of the preceding articles, 



Q = a \ f 2gh, 



(3) 



Art. 21.] HORIZONTAL ORIFICES. 4 1 

provided that the area of the orifice be small compared with 
the cross-section of the vessel. If a is in square feet and h in 
feet, Q will be expressed in cubic feet per second. It will be 
seen in the next chapter tnat various circumstances materially 
modify in practice the results as obtained from this formula. 

The discharge from a horizontal orifice is, like the velocity, 
proportional to the square root of the head. Thus with the 
same orifice to double the discharge requires the head to be 
increased fourfold' The head which will produce a given dis- 
charge is ^ 

h = 



2ga* 

whence the head varies inversely as the square of the area of 
the orifice. 

Horizontal orifices are but little used, as in practice it is 
found more convenient to arrange an opening in the side of a 
vessel than in the base. The above formula applies approxi- 
mately to a vertical orifice if h be taken as the head on its 
centre of gravity. 

The discharge is theoretically independent of the shape of 
the orifice, so that orifices of different forms with equal areas 
give the same value of Q. For a circle whose diameter is d, 

Q = Ind' 1 \/~2gk. 

For a rectangle whose sides are b and d, 
Q = bd 4/ w 2 . 

and similarly for other forms a is to be inserted in terms of the 
linear dimensions, which must be numerically expressed in the 
same unit as g. 

Prob. 29 Compute the theoretic discharge from an orifice 
one inch in diameter under a head of 1.5 feet. 



42 



THE ORE TIC A L H YDRA ULICS. 



[Chap. III. 



Article 22. Rectangular Vertical Orifices. 

If the size of an orifice in the side of a vessel be small com- 
pared with the head, then the mean theoretic velocity of the 
outflowing water may be taken as V2g/i, where h is the head on 
the centre of the orifice, and consequently the theoretic dis- 
charge is a V or a V2gh. Strictly, however, the head, and hence 

the velocity, is different in dif- 
ferent parts of an orifice whose 
plane is vertical. 

A rectangular orifice with 
two edges parallel to the water 
surface is the most important 
case. Let b be its breadth, /i l 
the head of water on its upper 
edge, and /i 2 the head on its 
lower edge, so that h 2 — /i 1 is its depth. Let any elementary 
strip whose area is b . Sy be drawn at a depth y below the water 
level. The velocity of flow through this elementary strip is, as 
shown in Art. 20, 

V= Vlgy, 

and the discharge per second through it is 





4. ■"' 

Hi 


* •■• — 


== 


; ■• )1T^ 


*2 




■*—*—» 




\ I 


7 




! 














< -77 ^ 







Fig. 



dQ = bdy V2gy. 

The total discharge through the orifice is obtained by integrat- 
ing this expression between the limits Ji x and k 3 , which gives 

Q = VV&W-kf) (9) 

In case the top edge of the orifice is at or above the level of 
the water, /i x — o, and then if the head h. 2 be denoted by H, 
the discharge is 

e = |j^i=iwi , p=|, ; ip . . ( 9 )' 

which is the basis of all formulas for weir measurement. 



Art. 22.] RECTANGULAR VERTICAL ORIFICES. 43 

To ascertain the error caused by using the formula (8) in- 
stead of (9) for a rectangular lateral orifice, let h be the head on 
its centre of gravity, and d pe its vertical depth, h 2 — h x . Then 
from (8) 

Q = bd V2gh. 

Now in (9) let /i 2 = h -f- id, and h x = h — id; then developing 
by the binomial formula, 

/ 3 d 3d' id 3 3__d' j$_ (P 

h? - k*\i + - 7i + j 2 tf~ Y5s¥ + To^s¥~ sig^¥ + etc - 
'._ ,./ U sd^ i_ c r ±JlsA^ d ls^ 

n? - n>\\ 4 /^ 32/^ 128 /Z" 1 " 2048/^ 8192/^ ~ t_etc " 
and (9) becomes 

.. ,—,[ id" id" id" x 

Q = bdV2gh[i - ~ 6¥ - — u , -—-- etc.). 

Therefore the discharge obtained by using (8) is always too 
great. The true theoretic discharge, from the formula just 
deduced, is: 

If h— d, Q = 0.989 bd Vzg/i; 

If h — 2d, Q = 0.997 bd V2gh ; 

If k = 3d, Q = 0.999 bd V2gh. 

The error of the formula Q = bdV^gk is thus seen to be 1.1 
per cent when h — d, only 0.3 per cent when h = 2d, and only 
about 0.1 per cent when h = 3d. Accordingly, if the head on 
the centre of the orifice is greater than two or three times the 
vertical depth of the orifice, the approximate formula (8) is 
generally used instead of the exact formula (9), since the slight 
error thus introduced is of no practical importance. 

Prob. 30. Compute the theoretic discharge from a rectan- 
gular orifice 0.5 feet wide and 0.25 feet high when the head on 
the top of the orifice is 0.375 f eet - 

Ans. Q —- 0.707 cubic feet per second. 



44 



THEORE 1 RCA L H YDRA ULICS. 



[Chap. III. 



^w*. 


— 

H 

t 





Article 23. Triangular Vertical Orifices. 

Triangular vertical orifices arc sometimes used for the 
measurement of water, the arrangement being as shown in 

Fig. 14. Let b be the width of 
the orifice at the water level, 
and H the head of water on 
the vertex. Let an elementary 
strip whose depth is dy be 
drawn at a distance/ below the 
water level. From similar triangles the length of this strip is 

~tt{H — y), and the elementary discharge then is 

dQ = —(H- y)dy V^gy = w V^g{Hy^ - j-)c5>. 
The integration of this between the limits o and H gives 



Fig. 



Q = T%b \>2gH- = &bH\ 2gH. 
If the sides of the triangle are equally inclined to the vertical, 
as should be the case in practice, and if this angle be a, b may 
be expressed in terms of a and H, so that the equation be- 
comes 



Q 



-^ tan a . H 2 V 2gH = -& tan a . V2g . HI 



The discharge is thus equal to a constant multiplied by the 
2.\ power of the measured depth. 

If the orifice be a trapezoid whose upper base is l\ lower 
base b ; , and altitude d, the discharge is found by integrating 
the above differential expression between the limits o and d, 
and then substituting for H its value in terms of d, b, and b', 

namely, H = 



b'' 

Q 



The theoretic discharge then is 



^- + i r)- 



Art. 24.] 



CIRCULAR VERTICAL ORIFICES. 



45 



If in this b' equals b it becomes the same as the formula for a 
rectangular orifice, while if b' equals o it gives the same result 
as found above for the triangle. 

Prob. 31. Prove that the theoretic discharge from a lateral 
triangular orifice whose base is horizontal and whose vertex is 
in the water level is Q = \bd V2gd, where b is the base and d\s> 
the altitude. 



Article 24. Circular Vertical Orifices. 

To determine the theoretic discharge through a circular 
orifice whose plane is vertical, let h be the head on its centre, 
and r its radius. Let an elementary- 
strip be drawn at a distance y above 
the centre ; the length of this is E 
2 Vr 2 — j/ 2 , its area is 2$y Vr* — y 2 , 
and the head upon it is h — y. Then 
the theoretic discharge through this 
strip is 

SQ = 2(5> Vr* — y 2 V 2 s (/i — y). FlG - J s. 

To integrate this expand (k — y)l by the binomial formula. 
Then it may be written 




<SQ=2 V2gh 



\™x=g*-<e=g£J£g£^-\* 



Each term of this expression is now integrable, and taking the 
limits of y as -\- r and — r the entire circle is covered, and 



1 r 



Q = n 1 >y 2g %l---- 



5 r 4 105 r 6 \ , v 

71 — 2 — ^y-a — etc -h (10) 
024/1' 65536/^ / \ J 



which gives the theoretic discharge per second for any values 
of r and h. 



4 6 



THE ORE TIC A L H YDRA ULICS. 



[Chap. III. 



The approximate formula (8) applied to this case gives for 
the discharge nr V 2g/i, which is always greater than the true 
discharge ; thus from (10), 

If h — 2r, Q = 0.992 nr 2 \ / 2gh; 
If h = $r, Q = 0.996 nr 2 V-2gh\ 
If h = 47', Q = 0.998 nr* \ / 2gh. 

Hence the error in the use of (8) is only 0.4 per cent when 
h = 3^, and only 0.2 per cent when h = A^r. In general the 
approximate formula may be used whenever the head on the 
centre of the circle is greater than four or five times its radius. 

Prob. 32. Compute the theoretic discharge from a circle of 
one inch diameter when the head on its centre is 0.5 feet. 



Article 25. Influence of Velocity of Approach. 

Thus far, in the determination of the theoretic velocity and 
discharge from an orifice, the head has been regarded as con- 
stant. But the head can only be maintained constant by an 
inflow of water, and this modifies the theoretic velocity. Let 
a be the area of the orifice, and A that of the horizontal cross- 
section of the reservoir ; let V be the theoretic velocity of flow- 
through a, and v the vertical velocity of inflow through the 
section A. The energy of W pounds of water as it flows from 

the orifice is W — , and this is equal to the energy Wh stored 

2 g 

up in the fall plus the energy n 7 _L G f the inflowing water, or 



V 2 



Wh 4- IV 



Now the quantity of water which flows through the section a 



Art. 25.] INFLUENCE OF VELOCITY OF APPROACH. 47 

in a unit of time is the same as that passing through the area 
A in the same time, or (Art. 19) 

aV '= Av, whence v = — V. 

A 

Inserting this value of v in the equation of energy, and solving 
for V, gives the result 



which is always greater than the value \2gh. 

The influence of a constantly maintained head on the ve- 
locity of flow at the orifice can now be ascertained by assign- 

ct, 

mg values to the ratio — thus: 
A 



If 


a— A, 


V= 00; 


If 


a= %A, 


V= 1.342 ^2p; 


If 


a = iA, 


V — 1.1 54 V2gh; 


If 


a= iA, 


V = 1. 06 1 V2gh\ 


If 


a= \A, 


V = 1 .02 1 V2g/i ; 


If 


a = T \A, 


V — 1.005 V2gh. 



It is here indicated that the common formula (8) is in error 
2.1 per cent when a = \A, if the head be maintained constant 
by a uniform vertical inflow at the water surface, and 0.5 per 
cent when a = -^A. Practically, if the area of the orifice be 
less than one-twentieth of the cross-section of the vessel, the 
error in using the formula V =■ V2gh is too small to be noticed 
even in the most precise experiments, and fortunately most 
orifices are smaller in relative size than this. 

A more common case is that where the reservoir is of large 



4 8 



THE ORE TIC A L H YDRA ULICS. 



[Chap. III. 



horizontal and small vertical cross-section, and where the water 

approaches the orifice with a 
horizontal velocity, as in a 
canal or trough. Here let A 
be the area of the vertical 
cross-section of the vessel, a 
the area of the orifice, and h 
Fig. 16. the head upon its centre. 

Then if h be large compared with the depth of the orifice, 

exactly the same reasoning applies as before, and the theoretic 

velocity of flow is 




V = 



igh 



i — 



If, however, h be small, let h x and k s be the heads on the upper 
and lower edges of the orifice, which is taken as rectangular. 
Then, using the same reasoning as above, the velocity of flow 
at any depth y is given by 

V'=2gy+v\ 

where v is the constant velocity of approach through the area 
A. The discharge through a strip of the length b and depth 
6y (Art. 20) then is 

dQ = bSy(2gy + v>f, 

and, by integration between the limits h x and h % , the theoretic 
discharge per second from the orifice is 



Q 



b V2.fr 



[(*■+ $'- (*• +!)']•• ■<"? 



In this case, particularly when //, = o, the velocity of approach 
may exercise a marked influence on the discharge. 



Art. 26.] * FLOW UNDER PRESSURE. 49 

Prob. 33. In the case of horizontal approach, as seen in 
Fig. 16, let b — 4 feet, // 2 = 0.8 feet, h x — o, and v = 2.5 feet 
per second. Compute the theoretic discharge : first, neglecting 
v ; and second, regarding v. 



Article 26. Flow under Pressure. 

The level of water in the reservoir and the orifice of out- 
flow have been thus far regarded as subjected to no pressure. 
or at least only to the pressure of the atmosphere which acts 
upon both with the same mean force of 14.7 pounds per square 
inch (since the head h is rarely or never so great that a 
sensible variation in atmospheric pressure can be detected 
between the orifice and the water level). But the upper level 
of the water may be subject to the pressure of steam or to the 
pressure due to a heavy weight or to a piston. The orifice 
may also be under a pressure greater or less than that of the 
atmosphere. It is required to determine the velocity of flow 
from the orifice under these conditions. 

First, suppose that the surface of the water in the vessel or 
reservoir is subjected to the uniform pressure of/ pounds per 
square foot above the atmospheric pressure, while the pressure 
at the orifice is the same as that of the atmosphere. Let h be 
the depth of water on the orifice. The velocity of flow Fis 
greater than ytgh on account of the pressure p Q , and it is 
evidently the same as that from a column of water whose 
height is such as to produce the same pressure at the orifice. 
The total unit-pressure at the depth of the orifice is 

p = w/i+p , 

and from (1) the head of water which would produce this pres- 
sure is 

w w 



50 THEORETICAL HYDRAULICS. |"Chap. IIL 

Accordingly the velocity of flow from the orifice is 



or, if /z denote the head corresponding to the pressure p , 



V=V2g(/l + k ). 

The general formula (6) thus applies to any small orifice, if h 
be the head corresponding to the static pressure at the orifice. 

Secondly, suppose that the surface of the water in the 
vessel is subjected to the unit-pressure p , while the orifice is 
under the external unit-pressure p v Let h be the head of 
actual water on the orifice, /i the head of water which will 
produce the pressure /„, and 1i 1 the head which will produce p x . 
The velocity of flow at the orifice is then the same as if the 
orifice were under a head h -\- /i — h x , or 



V- V2g(/l + h,-hZ, (12) 

in which the values of /i, and k, are 

h =£, h t =^. 

zu w 

Usually p, and/, are given in pounds per square inch, while h 
and /i x are required in feet ; then (Art. 9) 

/*„ = 2.304 A » 7/ r= 2.304 /,. 

The values of p and /, may be absolute pressures, or merely 
pressures above the atmosphere. In the latter case p x may 
sometimes be negative, as in the discharge of water into a 
condenser. 

As an illustration of these principles let a cylindrical reser- 



Art. 26.] 



FLOW UNDER PRESSURE. 



51 



voir, Fig. 17, be 2 feet in diameter, and upon the surface of the 
water let there be a tightly fitting 
piston which with the load W 
weighs 3000 pounds. At the 
depth 8 feet below the water 
level are three small orifices : one 
at A, upon which there is an ex- 
terior head of water of 3 feet; one 
not shown in the figure, which 
discharges directly into the at" 
mosphere ; and one at C, where the discharge is into a vessel in 
which the tension of the air is only 10 pounds per square inch. 
It is required to determine the velocity of efflux from each 
orifice. The head h corresponding to the pressure on the 
upper water surface is 




Fig. 17. 



h : ± A_ = 300 

" w 3.1416 X 62.5 



= 15.28 feet. 



The head h x is 3 feet for the first orifice, o for the second, and 
— 2.304(14.7— 10) = — 10.83 feet for the third. The three 
theoretic velocities of outflow then are : 



36. 1 feet per second ; 

38.7 feet per second ; 

V — 8.02 VS+ 15.28+ 10.83 = 46.8 feet per second. 



V= 8.02 VS+ 15.28 — 


3 


V= 8.02 VS+ 15.28 — 






In the case of discharge from an orifice under water, as at 
A in Fig. 17, the value of h — h x is the same wherever the 
orifice be placed below the lower level, and hence the velocity 
depends upon the difference of level of the two water surfaces, 
and not upon the depth of the orifice. 



The velocity of flow of oil or mercury under pressure is to 
be determined in the same manner as water, by finding the 



52 THEORETICAL HYDRAULICS. [Chap. IIL 

heads which will produce the given pressure. Thus in the pre- 
ceding numerical example, if the liquid be mercury, whose 
weight per cubic foot is 850 pounds, the head of mercury cor- 
responding to the pressure of the piston is 

3 000 

h a = - 7 „ — = I.X2 feet, 

3.1416x850 

and, accordingly, for discharge into the atmosphere at the 
depth h = 8 feet the velocity is 



V — 8.02 V84-1T12 = 24.2 feet per second, 

while for water the velocity was 38.7 feet per second. The 

general formula (6) is applicable to all cases of the flow of 

P 
liquids from a small orifice, if for h its value — be substituted, 

where/ is the resultant unit-pressure at the depth of the orifice, 
and w is the weight of a cubic unit of the liquid. 

Prob. 34. Water under a head of 230 feet flows into a boiler 
whose gauge reads 45 pounds per square inch. Find the ve- 
locity of the inflowing water. 

Prob. 35. The pressure in a boiler is 60 pounds per square 
inch above the atmosphere. Compute the theoretic velocity 
of flow from a small orifice one foot below the water level. 



Article 27. Pressure-head and Velocity-head. 

When a vessel is filled with water at rest the pressure at 
any point depends only upon the head of water above that 
point (Art. 9). But when the water is in motion it is a fact of 
observation that the pressure becomes less than that due to 
the head. The actual pressure in any event may be measured 
by the height of a column of water. Thus if the water be at 



Art. 27.] PRESSURE-HEAD AND VELOCITY-HEAD. 



53 




rest in the case shown in Fig. 18, and small tubes be inserted 

at A, B, and C, the water will 

rise in each tube to the same 

height as that of the water 

level in the reservoir, and the 

pressures at A, B, and C will 

be those due to the heads Aa t 

Bb, and Cc. But if an orifice 

be opened, as seen near C, the FlG - l8 - 

water levels in the tubes sink to the points a lt b x , and c x ; that 

is, the pressures at A, B, and C are reduced to those due to 

the heads Aa y , Bb, , and Cc x . 

Let h be the head of water on any point, or the depth of 
that point below the free water level. Let h x be the head 
due to the actual pressure of the water at that point, or the 

pressure-head. Let — be the head due to the actual velocity 

of the water at that point, or the velocity-head. Then 



4+ 



*g 



(13) 



or, in the form of a theorem : 

The pressure-head plus the velocity-head is equal to the 
total hydrostatic head. 

In order to prove this let W be the weight of water which 

passes the section per second ; then W — is the energy which 

<5 

it possesses. The total theoretic energy of this water is Wh, 
and if there be no losses of energy the remaining energy is 

W [h J , which is to be equated to Wh x , which represents 

the potential energy still existing in the form of pressure. 



54 THEORETICAL HYDRAULICS. [Chap. IIL 

Hence h — — = h. , 

2g 

whence the theorem follows as stated. In Fig. 18 aa x is the 
velocity-head for the section A, while Aa x is the pressure-head. 

Another method of proof is to consider the section at A as 
an orifice through which the flow occurs under a head h — h x , 
where h l is the head caused by the back pressure p x . Then, 
from the last article, 



v — V2g(k — h x \ 

v 1 
from which — = h — h. , which also agrees with the theorem. 

The pressure-head Aa 1 at A hence decreases when the ve- 
locity of the water at A increases, and the same is true for any 
other section as B. Let v and v' be the velocities at A and B ; 
then, since the same quantity of water passes each section per 
second, the relation Av — Bv' must be fulfilled. Hence if B 
be greater than A the velocity v is greater than v', and the 
pressure-head at B will be greater than at A. To illustrate : let 
the depths of A and B be 6 and 5 feet respectively below the 
water level, and the corresponding cross-sections be 1.2 and 2.4 
square feet. Let the quantity of water discharged by the 
orifice near C be 14.4 cubic feet per second. Then the velocity 
at A is 

14.4 

v = = 12 feet per second, 

1.2 r 

which corresponds to a velocity-head of 

v* 

— = 0.015557/ = 2.24 feet ; 

2g 

and accordingly the pressure-head Aa x is 

h x = 6.0 — 2.24 = 3.76 feet. 



Art. 27.] PRESSURE-HEAD AND VELOCITY HEAD. 



55 



Proceeding in the same way for B, the velocity is found to be 
6 feet per second, the velocity-head 0.56 feet, and finally the 
pressure-head is 5.0 •— 0.56 = 4.44 feet. The hydrostatic head 
at A is thus diminished by the velocity-head aa 1 = 2.24 feet, 
while at B it is diminished by the smaller amount bb 1 = 0.56 
feet. When the water was at rest the pressures were : 

At A, p = 0.434 X 6 = 2.60 pounds per square inch ; 
At B, p = 0.434 X 5 == 2.17 pounds per square inch. 

But as soon as the flow from the orifice began the pressures 
became: 

At A t p = O.434 X 3.76 = 1.63 pounds per square inch ; 
At B, p = 0.434 X 444 = 1.93 pounds per square inch. 



A negative pressure may occur if the velocity-head becomes 
greater than the hydrostatic head ; for since h x -\ equals h, 

the value of /z, is negative when — exceeds h. A case in 
which this may occur is shown in Fig. 19, where the section at 



A is so small that 



becomes 



larger than h, so that if a tube be 
inserted no water runs out, but if 
the tube be carried downward into 
a vessel of water there will be 
lifted a column CD whose height 
is that of the negative pressure- 
head h x . For example, let the 
cross-section of A be 0.4 square 
feet, and its head h be 4.1 feet, while 8 cubic feet per second 
are discharged from the orifice below. Then the velocity at A 
is 20 feet per second, and the corresponding velocity-head is 




Fig. 



$6 THEORETICAL HYDRAULICS. [Chap. III. 

6.22 feet. The pressure-head at A then is, from (13), 

h x = 4.1 — 6.22 = — 2.12 feet, 
and accordingly there exists at A an inward, or negative 
pressure, 

p x — — 2.12 X 0.434 — — °-9 2 pounds per square inch. 

This negative pressure will sustain a column of water CD 
whose height is 2.12 feet. If the small vessel be placed so that 
its water level is less than 2.12 feet below, water will be con- 
stantly drawn from the smaller to the larger vessel. This is 
the principle of the action of the injector-pump. 

Prob. 36. The hydrostatic pressure in a pipe is 80 pounds 
per square inch. What velocity must the water have to reduce 
this to 50 pounds per square inch? 

Article 28. Time of Emptying a Vessel. 

Let the depth of water in a vessel be H \ it is required to 
determine the time of emptying it through a small orifice in 
the base whose area is a. Let Y be the area 
of the water surface when the depth of water 
is y; let St be the time during which the water 
level falls the distance Sy. During this time 
the quantity of water Y6y passes through the 
Fig. 20. orifice. But the discharge in one second under 

the constant head y is a V2gy, and hence the discharge in the 
time 6t is aSt \ f 2gy. Equating these two expressions, there is 
found the relation 

YSy 




8t = 



a \ f 2gy 



The time of emptying the vessel is now found by inserting for 
Y its value in terms of y, and then integrating between the 
limits H and o. 



Art. 28.] TIME OF EMPTYING A VESSEL. 57 

For a cylinder or prism the cross-section Y has the constant 
value A, and the formula becomes 

a V 2g 
the integration of which gives 

, _ 2A s/H 2AH 



a V2g a V2gH 

as the theoretic time of emptying the vessel. If the head were 
maintained constant the uniform discharge per second would 
be a V2gH, and the time of discharging a quantity equal to 
the capacity of the vessel is AH divided by a V2gff, which is 
one half of the time required to empty it. 

To find the time of emptying a hemispherical bowl of 
radius r, let x be the radius of the cross-section Y; then 

x* = 2ry — j/ 2 ; 
Y = n(2ry — y). 
The equation for dt then becomes 

St = — -= (2/7* — 7 f )<5>, 
a \ 2g 

and by integration between the limits r and o 

I47rr f 



i$a V2g 
which is the theoretic time required to empty the hemisphere. 

The only important application of these principles is in the 
case of the right prism or cylinder, and the formula for this 
is materially modified in practice, as will be seen in the next 
chapter. It is more frequently required to determine the 
time during which the water level will descend from the 



58 THEORETICAL HYDRAULICS. [Chap. III. 

height H to another height h. This is found by integrating- 
between the limits H and h ; thus, for the prismatic vessel, 

2A 
t= ^W^^ • • • • • (H) 

which gives the theoretic time of descent in seconds. 

Prob. 37. A sphere is filled with water. Find the time of 
emptying it through a small orifice at its lowest point. 

Prob. 38. A conical vessel whose altitude is H, and whose 
base has the radius r, is placed with its axis vertical, and 
emptied through a small orifice in its base. Prove that the 

theoretic time is 7=~* 

i$a \2g 

Article 29. Flow from a Revolving Vessel. 

The water in a vessel at rest is acted upon only by the 
force of gravity, and hence its surface is a horizontal plane ; but 
the water in a revolving vessel is acted upon by a centrifugal 
force as well as by gravity, so that its surface assumes a curved 
shape. The simplest case is that of a vessel revolving with 
uniform velocity about a vertical axis, and it will be shown 
that here the water surface forms a paraboloid whose axis 
coincides with that about which it revolves. Fig. 21 repre- 
sents such a case, NT being the vertical axis. 

Let M be any point on the surface whose co-ordinates ON 
and NM are y and x. Let IV be the 
weight of a particle at 31, whose intensity 
is represented by MG ; this particle in 
consequence of its velocity of revolution 
u is acted upon also by a centrifugal force 

MC whose value* is — The resultant 

g x 

See Wood's Elementary Mechanics, p. 226. 




Art. 29.] FLOW FROM A REVOLVING VESSEL. $9 

MR of the weight and centrifugal force must be normal to the 
tangent MS at M, as the condition of equilibrium. The angle 
NMS is hence equal to RMG, and accordingly 

UnMMS = ^ = ^. 
M& gx 

But the tangent of this angle is the first derivative of y with 
reference to x. Further, the value of u varies directly with 
x, so that u = gox if go be the angular velocity, that is, the 
velocity at the distance unity from the axis. Accordingly, 

6y _ u 2 _ gq* 

dx ~ gx ~ g 

is the differential equation of the curve, and by integration 

2 2 

GO X 



which is the equation of a common parabola. Therefore the 
surface is a paraboloid. Since gox is the velocity u at the point 
My this equation may be written 

if 

y — — f 
*g 

which shows that the ordinate y is the head due to the velocity 
of revolution. 

If h be the head O T at the axis, the velocity of efflux 
from a small orifice at T is \2gh. But for an orifice at £7 the 
velocity is due to the head MU, and 

MU= OT + NO = fi+y. 
The theoretic velocity of flow from U therefore is 



V = V2g(k +y)=- V2gh + u% .... (1 5) 

where u is the velocity of revolution of the point U or M. 
This formula is a very important one in the discussion of cer- 
tain hydraulic motors. 



60 THEORETICAL HYDRAULICS. [Chap. III. 

To determine the velocity u of a point at the distance x 
from the axis of revolution it is only necessary to count the 
number of revolutions made per second. If n be this number, 

u = 2nx . n ; 

or, in another form, since 2nn is the velocity at the distance 
unity from the axis, 

go = 27i7t and u = cox. 

As an example of the application of these principles, let 
there be a cylindrical vessel which is 2 feet in diameter and 
3 feet deep, and which is one half full of water. It is required 
to find the number of revolutions per second about its axis 
which will cause the water to begin to overflow around the 
upper edge. The volume of a paraboloid being one-half of 
its circumscribing cylinder, the vertex of the paraboloid at the 
moment of overflow will coincide with the centre of the base 
of the vessel, and hence the value of y for the upper edge is 
3 feet. Accordingly, 



J> = 3 = 



*g 

whence go = 13.89, and then 

13.89 



n == 



27t 



which is the number of revolutions per second. If the vessel 
were three-fourths full of water, the volume of the paraboloid 
at the moment of overflow would be one-fourth that of the 
cylinder, and the value of y for the upper edge would be one- 
half the altitude of the cylinder, or 1.5 feet. Hence go is found 
to be 9.82, whence the number of revolutions per second is 
about 1.56. 

Prob. 39. A cylindrical vessel is 3 feet in diameter. How 
many revolutions per minute must be made about its vertical 



Art. 30.] 



THE PATH OF A JET. 



6l 



axis in order that the velocity of the outer surface may be 50 
feet per second ? 

Prob. 40. A cylindrical vessel 2 feet in diameter and 3 feet 
deep is three-fourths full of water, and is revolved about its 
vertical axis so that the water is just on the point of overflow- 
ing around the upper edge. Find the theoretic velocity of 
efflux from an orifice in the base at a distance of 9 inches from 
the axis Ans. 12.28 feet per second. 

Article 30. The Path of a Jet. 

When a jet of water issues from a small orifice in the ver- 
tical side of a vessel or reservoir, its di- 
rection at first is horizontal, but the 
force of gravity immediately causes the 
jet to move in a curve which will be 
shown to be the common parabola. 
Let x be the abscissa and y the ordi- 
nate of any point of the curve, meas- 
ured from the orifice as an origin, as 
seen in Fig 22. The effect of the im- 
pulse at the orifice is to cause the space x to be described 
uniformly in a certain time t> or, if v be the velocity of flow, 
x == vt. The effect of the force of gravity is to cause the 
space y to be described in accordance with the laws of falling 
bodies (Art. 6), or y = \gf. Eliminating t from these two 
equations gives 

y 





— k — k=* 


X / 

1 — Z^*" 

\ St 


— \i- 

---l- 
1 


— wtfdL — w*y . 


1 

1 

1 



Fig. 22. 



gx_ 

2V* 



X 
4A 9 



which is the equation of a parabola whose axis is vertical and 
whose vertex is at the office. 

The horizontal range of the jet for any given ordinate y 
is found from the equation x 2 = 4/iy. If the height of the 
vessel be /, the horizontal range on the plane of the base is 



x = 2 Vk(l - A). 



62 



THE ORE TIC A L HYD RA ULICS. 



[Chap. III. 



This value is o when h = o and also when h = /, and it is a 
maximum when h = \l. Hence the greatest range is from an 
orifice at the mid-height of the vessel. 

A more general case is that where the side of the vessel is 

inclined to the vertical at the 
' angle 8, as in Fig. 23. Here the 

fi= jet at first issues perpendicularly 
to the side, and under the action 
of the impulsive force a particle 
of water would describe the dis- 
tance AB in a certain time t. 
IG ' 23 " But in that same time the force 

of gravity causes it to descend through the distance BC. Xow 
let x be the horizontal abscissa and y the vertical ordinate of 
the point C measured from the origin A. Then AB = x sec 8, 
and BC = x tan — y. Hence 




x sec 8 = vt, x tan 



y 



W-. 



The elimination of t from these expressions gives, after replac- 
ing v 2 by its value 2gh, 



x sec 

y = x tan 6 ; — , . . 

4/1 

which is also the equation of a common parabola. 



(16) 



To find the horizontal range in the level of the orifice make 
y .== o .\ then 

tan 6 



x = 4/1 



sec" 



2/1 sin 28. 



This is o when 8 =0° or 8 = 90°; it is a maximum and equal to 
2/1 when 8 = 45 °. To find the highest point of the jet the 
first derivative ofy with reference to x is to be equated to zero 



Art. 30.] THE PATH OF A JET. 63 

in order to locate the point where the tangent to the curve is 
horizontal ; thus, 

<5V x sec 2 6 

-f- = tan 6 -; — = o, 

dx 2k 

from which x = 2/1 sin 6 cos 6, and this, inserted in the equation 
■ of the curve, gives 

y = h sin 2 0, 

which is the highest elevation of the jet above the orifice. In 
this, if 6 == 90 , y — Ji ; that is, if a jet be directed vertically up- 
ward it will, theoretically, rise to the height of the level of 
water in the reservoir. 

As a numerical example let a vessel whose height is 16 feet 
stand upon a horizontal plane DE, Fig. 23, the side of the 
vessel being inclined to the vertical at the angle = 30 . Let 
a jet issue from a small orifice at A, under a head of 10 
feet. The jet rises to its maximum height, y = Jio = 2.5 feet, 
.at the distance x = \ 1/-3 X 10 .= 8.66 feet from A. At x — 
17.32 feet the jet crosses the horizontal plane through the 
orifice. To find the point where it strikes the plane DE, the 
value of y is made — 6 feet ; then, from the equation of the 
■curve, 

-6 = xVi , 

30 

from which x is found to be 24.62 feet ; whence, finally, 
DE = 24.62 — 6 tan 30 == 21.16 feet. 

Prob. 41. Find all the circumstances of the motion of a jet 
which issues from a vessel under a head of 5 feet, the side of 
the vessel being inclined to the vertical at an angle of 6o°, and 
its depth being 9 feet. 



64 THEORETICAL HYDRAULICS. [Chap. IIL 



Article 31. The Energy of a Jet. 

Let a jet or stream of water have the velocity z>, and let W 
be the weight of water per second passing any given cross- 
section. The energy of this moving water, or the work which 
it is capable of doing, is the same as that stored up by a body 
falling freely under the action of gravity through a height h 
and thereby acquiring the velocity v. Thus, if K be the energy 
or potential work, 

K= Wh= W— (17) 

Therefore, for a constant quantity of water per second passing 
through the given cross-section, the energy of the jet is pro- 
portional to the square of its velocity. 

The weight IF, however, may be expressed in terms of the 
cross-section of the jet and its velocity. Thus, if a be the area 
of the cross-section, and w the weight of a cubic unit of water, 
IV is the weight of a column of water whose length is v and 
whose cross-section is a, or W=wav\ and hence (17) maybe 
written 

K=~ (17/ 

ig 

In general, then, it may be stated that for a constant cross- 
section, the energy of a jet, or the work which it is capable of 
doing per second, varies with the cube of its velocity. 

The expressions just deduced give the theoretic energy of 
the jet, that is, the maximum work which can be obtained from 
it ; but this in practice can never be fully utilized. The amount 
of work which is realized when a jet strikes a moving surface, 



Art. 31.] THE ENERGY OF A JET. 65 

like the vane of a water-motor, depends upon a number of cir- 
cumstances which will be explained in a later chapter, and it 
is the constant aim of inveritors so to arrange the conditions 
that the actual work may be as near to the theoretic energy as 
possible. The " efficiency " of an apparatus for utilizing the 
energy of moving water is the ratio of the work actually 
utilized to the theoretic work ; or, if k be the work realized, 
the efficiency e is 

e=- (18) 

K 

The greatest possible value of e is unity, but this can never be 
attained, owing to the imperfections of the apparatus and the 
hurtful resistances. Values greater than 0.90 have, however, 
been obtained ; that is, 90 per cent or more of the theoretic 
work has been utilized in some of the best forms of hydraulic 
motors. 

For example, let water issue from a pipe 2 inches in diam- 
eter with a velocity of 10 feet per second. The cross-section 

3.1416 

111 square feet is , and the theoretic work in foot-pounds 

144 

per second is 

K— 0.01555 X 62.5 X 0.0218 X io 3 = 21.2, 

which is O.0385 horse-powers. If the velocity is 100 feet per 
second, however, the theoretic horse-power of the stream will 
be 38.5. 

Prob. 42. One cubic foot of water per second flows from an 
orifice with a velocity of 32 feet per second. Find the theo- 
retic horse-power of the stream. 

Prob. 43. A small turbine wheel using 102 cubic feet of 
water per minute under a head of 40 feet is found to give 6.i£ 
horse-power. Find the efficiency of the wheel. 

Ans. 80 per cent. 



66 THEORETICAL HYDRAULICS. [Chap. III. 

Article 32. The Impulse and Reaction of a Jet. 

When a stream or jet is in motion delivering W pounds of 
water per second with the uniform velocity v, that motion may 
be regarded as produced by a constant impulsive force F, which 
has acted upon IV for one second and then ceased. In this 
second the velocity of F has increased from o to v, and the 
space \v has been described. Consequently the work F X \v 
has been imparted to the water by the impulse/ 7 . But the 

theoretic energy of the jet is W — ; hence 

Fxiv= W- , 
from which the force of impulse F is 

F=W- (19) 

g 

Let a be the area of the cross-section of the jet ; then IV = wav, 
and 

F — wa — i iq) 

g 

Therefore the impulse of a jet of constant cross-section varies 
as the square of its velocity. 

The force F is a continuous impulsive pressure acting in 
the direction of the motion. For, by the definition, F acts for 
one second upon the W pounds of water which pass a given 
section ; but in the next second IF pounds also pass the section, 
and the same is the case for each second following. This im- 
pulse will be exerted as a pressure upon any surface placed in 
the path of the jet. 

The reaction of a jet upon a vessel occurs when water flows 
from an orifice. This reaction must be equal in value and 
opposite in direction to the impulse, as in all cases of stress 



Art. 32.] THE IMPULSE AND REACTION OF A JET, 6? 

action and reaction are equal. In the direction of the jet the 
impulse produces motion, in the opposite direction it produces 
a pressure which tends to move the vessel. The force of reac- 
tion of a jet hence is 

F = W — = wa — 
g g 

To compare this with hydrostatic pressure, let h be the ve- 
locity-head due to v ; then 

F = 2wa — = 2wah. 
*g 

But, from Art. 10, the normal pressure on a surface of area a 
under the hydrostatic head h is zvaJi. Therefore the dynamic 
pressure caused by the reaction of a jet issuing from an orifice 
in a vessel is double the hydrostatic pressure on the orifice 
when closed. This theoretic conclusion has been verified by 
experiment. 

The full force of impulse or reaction is exerted in the line 
of the action of the jet, and its force in any other direction is 
the component of the force F in that direction. Hence in a 
direction which makes an angle 6 with the line of motion of 
the jet, the force which can be exerted by the impulse or reac- 
tion is .Fcos 6. Thus if water issues from an orifice in the base 
of a vessel, it exerts an upward reaction F and a horizontal 
reaction o ; if it issues in a direction inclined 30 to the vertical, 
its upward reaction isi 7 cos3o°, and its horizontal reaction is 
i^sin 30 . 

If a stream moving with the velocity v 1 is retarded so that 
its velocity becomes v 2 , its impulse in the first instance is 

W -, and in the second W-. The difference of these, or 

g 

g 



68 THEORETICAL HYDRAULICS. [Chap. III. 

is a measure of the dynamic pressure developed. It is by 
virtue of the pressure due to change of velocity that turbine 
wheels and other hydraulic motors transform the energy of 
moving water into useful work. 

Prob. 44. Devise an experiment for measuring the force of 
reaction of a jet which issues from an orifice in the base or side 
of a vessel. 



Article 33. Absolute and Relative Velocities. 

Absolute velocity is that with respect to the earth, and 
relative velocity that with respect to a body in motion. For 
instance, if water issues from a small orifice in a vessel which is 
in motion in a straight line with the uniform velocity u, the 
theoretic velocity of flow relative to the vessel is V= \ 2gk, 
or the same as its absolute velocity if the vessel were at rest, 
V v for no accelerating forces exist to 

change the direction or the value 
of g. The absolute velocity of 
flow, however, may be greater or 
less than V, depending upon the 
value of it and its direction. To 
illustrate : Fig. 24 shows a moving vessel from which water is 
flowing through three orifices. At A the direction of V is 
horizontal, and as the vessel is moving in the opposite direction 
with the velocity ?/, the absolute velocity of the water as it 
leaves the orifice is 

. V = V — 21. 

It is plain that if the orifice were in the front of the vessel and 
the direction of Fwere horizontal, the absolute velocity would 
be v — V-\-u. 

Again, at B is an orifice from which the water issues verti- 
cally with respect to the vessel with the relative velocity V t 




Art. 33-] ABSOLUTE AND RELATIVE VELOCITIES. 6g 

while at the same time the orifice moves horizontally with the 
velocity u. Forming the parallelogram, the absolute velocity 
v is seen to be the resultant of Fand ic, or 



Lastly, at C is shown an orifice in the front of the vessel so 
arranged that the direction of the relative velocity V makes an 
angle <fi with the horizontal. From C draw Cu to represent 
the velocity u, and CV to represent V, and complete the par- 
allelogram as shown ; then Cv, the resultant of u and V, is the 
absolute velocity with which the water leaves the orifice. 
From the triangle Cuv, 



v — Vif -[- V 2 -f 2u Fcos (p. 

In this, if <p = o, v becomes u -f- V as before shown ; if = 90 , 
it becomes the same as when the water issues vertically from 
the orifice in the base; and if <fi — 180 , the value of v is that 
before found for an orifice in the side of the vessel. 

In Art. 29 the velocity of flow from an orifice in a vessel 
revolving with uniform velocity was found to be 



V = V2gh + u\ 

This is the velocity relative to the vessel. If the orifice be in 
the base, the direction of V with respect to the vessel is ver- 
tical, and as the orifice is moving horizontally with the uniform 
velocity u, the absolute velocity of flow is 



v = Vu 1 + V 2 = V2gh + 2u\ 

In the same way, if the orifice be in the side of the vessel, and 
the direction of V be horizontal and directly away from the 
axis, the same formula applies, for the absolute velocity v is 
the resultant of the two rectangular components Fand 21. 



70 THEORETICAL HYDRAULICS. [Chap. III. 

If a vessel move with a motion which is accelerated or re- 
tarded, this affects the value of g, and the reasoning of the pre- 
ceding articles does not give the correct value of V. For 
instance, if a vessel move vertically upward with an accelera- 
tion fi the theoretic relative velocity of flow from an orifice in 
it is 

V=V2{g-\-f)k; 

and if it be its velocity at any instant, the absolute velocity 
of flow is u -f- V. This equation shows that if a vessel be 
moving downward with the acceleration g, that is, freely 
falling, Fwill be zero, which of course is to be expected since 
both water and vessel are alike accelerated. 

Prob. 45. If V be velocity of flow from the orifice at A in 
Fig. 23, show that the velocity of the jet at the point E 

is W 2 + 2gH. 

Prob. 46. If a vessel of water is moving horizontally with 
an acceleration \g, show that the surface of the water is a 
plane which is inclined to the horizontal at an angle of about 
14 degrees. 



Art. 34.] 



THE STANDARD ORIFICE. 



71 



CHAPTER IV. 
FLOW OF WATER THROUGH ORIFICES. 

Article 34. The Standard Orifice. 

Orifices for the measurement of water are usually placed in 
the vertical side of a vessel or reservoir, but may also be placed 
in the base. In the former case it is understood that the 
upper edge of the opening is completely covered with water; 
and generally the head of water on an orifice is at least three 
or four times its vertical height. The term standard orifice 
is here used to signify that the opening is so arranged that 
the water in flowing from it touches only a line, as would 
be the case in a plate of no thickness. To secure this result 
the inner edge of the opening has a square corner, which alone 
is touched by the water. In precise experiments the orifice 
may be in a metallic plate whose 
thickness is really small, as at A in 
Fig. 25, but more commonly it is 
cut in a board or plank, care being 
taken that the inner edge is a 
definite corner. It is usual to bevel 
the outer edges of the orifice as at 
C, so that the escaping jet may by 
no possibility touch the edges ex= 
cept at the inner corner. The term " orifice in a thin plate " is 
often used to express the condition that the water shall only 
touch the edges of the opening along a line. This arrange- 




FlG. 



72 FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

ment may be regarded as a kind of standard apparatus for the 
measurement of water, for, as will be seen later, the discharge 
is modified if the inner corner is rounded, and different de- 
grees of rounding give different discharges. Orifices arranged 
as in Fig. 25 are accordingly always used when water is to be 
measured by the use of orifices. 

The contraction of the jet which is always observed when 
water issues from a standard orifice as described above is a 
most interesting and important phenomenon. It is due to the 
circumstance that the particles of water as they approach the 
orifice move in converging directions, and that these directions 
continue to converge for a short distance beyond the plane of 
the orifice. It is this contraction of the jet that causes only 
the inner corner of the orifice to be touched by the escaping 
water. The appearance of such a jet under steady flow, issuing 
from a circular orifice, is that of a clear crystal bar whose 
beauty excites the admiration of every observer. The place 
of greatest contraction is at a distance from the plane of the 
orifice of about one-half its diameter, and beyond this point 
the jet gradually enlarges in size, while its surface becomes 
more or less disturbed owing to the resistance of the air and 
other causes. In the case of square and rectangular orifices 
the contraction of the jet is also observed, its edges being 
angular and its cross-section similar to that of the orifice until 
the place of greatest contraction is passed. 

Owing to this contraction the discharge from a standard 
orifice is always less than the theoretic discharge. It is the 
object of this chapter to determine how the theoretic formulas 
are to be modified so that they may be used for the practical 
purposes of the measurement of water. This is to be done by 
the discussion of the results of experiments. It will be sup- 
posed, unless otherwise stated, that the size of the orifice is 
small compared with the cross-section of the reservoir, so that 



Art. 35.] THE COEFFICIENT OF CONTRACTION. 73 

the effect of velocity of approach may be neglected (Art. 25). 

Prob. 47. Under a head of 6 feet the discharge from an 
orifice is 3.74 gallons per second. What head is necessary in 
order that the discharge may be one cubic foot per second ? 

Article 35. The Coefficient of Contraction. 

The coefficient of contraction is the number by which the 
area of the orifice is to be multiplied in order to give the area 
of the least cross-section of the jet. Thus, if c' be the co- 
efficient of contraction, a the area of the orifice, and a! that of 
the jet, 

a' = c 'a (20) 

The coefficient of contraction is evidently always less than 
unity. 

The only direct method of rinding the value of c' is to 
measure by callipers the dimensions of the least cross-section 
of the jet. The size of the orifice can usually be determined 
with accuracy, but no great precision can be attained in 
measuring the jet. To find c' for a circular orifice let d and d' 
be the diameters of the sections a and a' ; then 

. d_ fdV 
a~ \~d 



(2oy 



Therefore the coefficient of contraction is the square of the 
ratio of the diameter of the jet to that of the orifice. In this 
way NEWTON found for c' the value 0.71 ; BORDA, 0.65 ; Bos- 
SUT, from 0.66 to 0.67; MlCHELOTTI, from 0.57 to 0.624 with 
a mean of 0.61. Eytelwein gave 0.64 as a mean value, and 
WEISBACH mentions 0.63. 

As a mean value the following may be kept in mind by the 
student : 

Coefficient of contraction c' = 0.62 ; 



74 FLO W OF WA TER THRO UGH ORIFICES. [Chap. IV. 

or, in other words, the minimum cross-section of the jet is 62 
per cent of that of the orifice. This value, however, undoubt- 
edly varies for different forms of orifices and for the same 
orifice under different heads, but little is known regarding the 
extent of these variations or the laws that govern them. Prob- 
ably c' is slightly smaller for circles than for squares, and 
smaller for squares than for rectangles, particularly if the rect- 
angle be long compared with its width. Probably also c' is 
larger for low heads than for high heads. 

Prob. 48. The diameter of a circular orifice is 1.995 inches. 
Three measurements of the diameter of the least cross-section 
of the jet give the values 1.55, 1.56, and 1.59 inches. Find the 
coefficient of contraction. 

Article 36. The Coefficient of Velocity. 

The coefficient of velocity is the number by which the theo- 
retic velocity of flow from the orifice is to be multiplied in 
order to give the actual velocity at the least cross-section of 
the jet. Thus, if c 1 be the coefficient of velocity, V the theo- 
retic velocity due to the head on the centre of the orifice, and 
v the actual velocity at the contracted section, 

V = C 1 V=C i \/2gh (21) 

The coefficient of velocity must be less than unity, since the 
force of gravity cannot generate a greater velocity than that 
due to the head. 

The velocity of flow at the contracted section of the jet 
cannot be directly measured. To obtain the value of the co- 
efficient of velocity, indirect observations have been taken on 
the path of the jet. Referring to Art. 30, it will be seen that 
when a jet flows from an orifice in the vertical side of a vessel 
it takes a path whose equation is 

*** 



Art. 36.] THE COEFFICIENT OF VELOCITY. 75 

in which x and y are the co-ordinates of any point of the path 
measured from vertical and horizontal axes, and v is the ve- 
locity at the origin. Now placing for v its value c 1 V 2gk, and 

solving for c 1 , gives 

x 

2 Vhy 

Therefore c x becomes known by the measurement of the two 
co-ordinates x and y and the head h. 

In conducting this experiment it would be well to have a 
ring, a little larger than the jet, supported by a stiff frame 
which can be moved until the jet passes through the ring. 
The flow of water can then be stopped, and the co-ordinates of 
the centre of the ring determined. By placing the ring at 
different points of the path different sets of co-ordinates can be 
obtained. The value of x should be measured from the con- 
tracted section rather than from the orifice, since v is the 
velocity at the former point and not at the latter. 

By this method of the jet BOSSUT in two experiments 
found for the coefficient of velocity the values 0.974 and 0.980, 
MlCHELOTTI in three experiments obtained 0.993, 0.998, and 
O.983, and Weisbach deduced 0.978. Great precision cannot 
be obtained in these determinations, nor indeed is it necessary 
for the purposes of hydraulic investigation that c 1 should be 
accurately known for standard orifices. As a mean value the 
following may be kept in the memory : 

Coefficient of velocity c x = 0.98 ; 

or, the actual velocity of flow at the contracted section is 98 
per cent of the theoretic velocity. The value of c 1 is greater 
for high than for low heads, and may probably often exceed 
0.99. 

Another method of finding the coefficient c x is to place the 
orifice horizontal so that the jet will be directed vertically up- 



76 FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

warn as in Fig. 12, Art. 20. The height to which it rises is the 
velocity height h , or 

v 2 



in which v is the actual velocity c x V2gh. Substituting, this 

value of v gives 

k e = c{h, 

from which, when h is measured, c x is computed. For ex- 
ample, under a head of 23 feet a stream was found to rise to a 
height of 22 feet ; then 

This method, like the preceding, fails to give good results for 
high velocities owing to the resistance of the air, and moreover 
it is impossible to measure with precision the height h a . 

Prob. 49. MlCHELOTTI found the range of a jet to be 6.25 
meters on a horizontal plane 1.41 meters below the vertical 
orifice, which was under a head of 7.19 meters. Compute the 
coefficient of velocity. 

Article 37. The Coefficient of Discharge. 

The coefficient of discharge is the number by which the 
theoretic discharge is to be multiplied in order to obtain the 
actual discharge. Thus, if c be the coefficient of discharge, 
Q the theoretical and q the actual discharge per second, 

q = cQ . (22) 

Evidently c is a number less than unity. 

The coefficient of discharge can be accurately found by 
allowing the flow from an orifice to fall into a vessel whose 
cubic contents are known with precision. The quantity q is 



Art. 37.] THE COEFFICIENT OF DISCHARGE. 7J 

thus determined, while Q is computed from the formulas of the 
last chapter. Then 

C- f (22)' 

For example, a circular orifice of 0.1 feet diameter was kept 
under a constant head of 4.677 feet; during a time of 5 minutes 
32-J- seconds the jet flowed into a measuring vessel which was 
found to contain 27.28 cubic feet. Here the actual discharge 
per second was 

q — v = 0.08212 cubic feet. 
332.2 

The theoretic discharge, from formula (8), is 



Q = n x 0.05 2 X 8.02 V 4.677 = 0.1 361 cubic feet. 

Then, for the coefficient of discharge, 

0.08212 ^ 

c = — - = 0.004. 

0.1361 

In this manner thousands of experiments have been made 
upon different forms of orifices under different heads, for ac- 
curate knowledge regarding this coefficient is of great impor- 
tance in practical hydraulic work. 

The following articles contain values of the coefficient of 
discharge for different kinds of orifices, and it will be seen 
that in general c is greater for low heads than for high heads, 
greater for rectangles than for squares, and greater for squares 
than for circles. Its value ranges from 0.59 to 0.63 or higher, 
and as a mean to be kept in mind by the student there may 
be stated : 

Coefficient of discharge c = 0.61 ; 



JS FLOW OF WATER THROUGH ORIFICES. [Chap IV, 

or, the actual discharge from orifices such as are shown in Fig. 
25 is 61 per cent of the theoretic discharge. 

The coefficient c may be expressed in terms of the coef- 
ficients c' and c 1 . Let a and a be the areas of the orifice and 
the cross-section of the contracted jet, and Q and q the theo- 
retic and actual discharge per second. Then 

q a c x \ 2g1i a' 
Q a V~2g/i a 

But (Art. 34) the ratio a' : a is the coefficient c' ; therefore 

c - c'c^ ; (23) 

or,, the coefficient of discharge is the product of the coefficients 
of contraction and velocity. 

Prob. 50. What is the discharge in gallons per minute from 
a circular orifice one inch in diameter under a head of 12 
inches, the coefficient of discharge being 0.609? 

Prob. 51. The diameter of a contracted circular jet was 
found to be 0.79 inches, the diameter of the orifice being 
one inch. Under a head of 4 feet the actual discharge per 
minute was found to be 3.21 cubic feet. Find the coefficient 
of velocity. 

Article 38. Circular Vertical Orifices. 

Let h be the head on the centre of a vertical circular orifice 
whose diameter is d. The theoretic discharge per second is 
found from formula (10), Art. 24, by placing for r its value ^d, 
and the actual discharge per second is 



q — c. \nct V2gh[ I 



I d\ 

128 /r 


5 <? 
16384** 




105 d" 



etc. , . (24) 
194 304 h I 



CIRCULAR VERTICAL ORLFICES. 



79 



Art. 38.] 

in which c is the coefficient of discharge. In case h becomes 
large compared with d, the negative terms in the parenthesis 
may be neglected, and 

q — c.\nd' 1 V 2g/i, (24') 

which is the same as the formula for horizontal circular orifices 
(Art. 21). 

The following table of values of c is abridged from the 
results deduced by Hamilton Smith, Jr.,* as determined by 
the discussion of all the best experiments. The table applies 
only to standard orifices. 



TABLE VI. COEFFICIENTS FOR CIRCULAR VERTICAL ORIFICES. 



Head 

h 
in Feet. 






Diameter of Orifice in 


Feet. 






0.02 


0.04 


0.07 


0.1 


0.2 


0.6 


1.0 


O.4 




0.637 


O.624 


0.618 








0.6 

0.8 


0.655 
.648 


.630 
.626 


.618 
•615 


.613 


601 


o.593 
•594 


590 


.610 


601 


1.0 


.644 


.623 


.612 


.608 


600 


•595 


591 


i-5 


.637 


.618 


.608 


.605 


600 


•596 


593 


2. 

2-5 


.632 
.629 


.614 
.612 


.607 
•605 


.604 
.603 


599 
599 


•597 


595 
596 


.593 


3- 
4 


.627 
.623 


.611 
.609 


.604 
•603 


.603 
.602 


599 
599 


.59S 


597 


597 


596 


6. 


.618 


.607 


.602 


.600 


59S 


•597 


59 6 


8. 


.614 


.605 


.601 


.600 


593 


• 596 


596 


10. 


.611 


.603 


•599 


•593 


597 


.596 


595 


20. 


.601 


•599 


•597 


.596 


59 6 


• 596 


594 


50. 


•59 6 


•595 


•594 


•594 


594 


•594 


593 


100. 


•593 


.592 


•592 


•59 2 


592 


592 


592 



This table shows that the coefficient c decreases as the size 
of the orifice increases, and that for diameters less than 0.2 



* Hydraulics, p. 59. 



80 FLOW OF WATER THROUGH ORIFICES. [Chap. IV; 

feet it decreases as the head increases. It may be presumed 
that the cause of this variation is due to a more perfect con- 
traction of the jet for large heads and large orifices than for 
small heads and small orifices. 

In applying the above coefficients to actual problems, the 
approximate formula 

q — c. iTtd 2 \ / 2gh 

may be used except for the values found above the horizontal 
lines in the last three columns. For these, if precision be re- 
quired, the accurate expression for q must be employed. The 
error committed by using the approximate formula for the 
values above the horizontal lines will depend upon the ratio of 
d to h ; as shown, in Art. 24, this error will be about two-tenths 
of one per cent when h =*2d, and about eight-tenths of one 
per cent when h = d. 

Prob. 52. Find from the table the coefficient of discharge 
for a circular orifice of two inches diameter under a head of 
1.75 feet. 

Prob. 53. Compute the probable actual discharge through a 
circular orifice of J inches diameter under a head of I foot 3 
inches. 

Article 39. Square Vertical Orifices. 

Let a square orifice whose side is d be placed with its edges 
truly parallel and perpendicular to a horizontal plane. Let h Y , 
/i 2 , and h be the heads of water on its upper edge, lower edge, 
and centre, respectively. The theoretic discharge per second 
is found by replacing b by d in formula (9) of Art. 22, and the 
actual discharge is 

q=c.ld«Tg{h}-h?) (25) 

Further, as shown in Art. 22, if // be large compared with d, 
the discharge may be computed by the simpler formula 

q = c.d*V^~h (25O 

In both formulas c is the coefficient of discharge (Art. 36). 



Art. 39.] 



SQUARE VERTICAL ORIFICES. 



81 



The following values of the coefficient c have been taken 
from a more extended table deduced by SMITH by an ex- 
haustive discussion of experiments. They are applicable only 
to cases where the orifice has a sharp inner edge so that the 
contraction of the jet may be perfectly formed (Art. 33). 

TABLE VII. COEFFICIENTS FOR SQUARE VERTICAL ORIFICES. 



Head 
h 

in Feet. 






Side of the Square 


in Feet. 






0.02 ■ 


0.04 


0.07 


O.I 


0.2 


0.6 


1.0 


0.4 




0.643 


O.628 


0.621 








0.6 


0.660 


.636 


• 623 


.617 


O.605 


O.598 




0.8 


.652 


.631 


.620 


.615 


.605 


.600 


0.597 


I .O 


.648 


.628 


.618 


•613 


• 605 


.601 


•599 


1-5 


.641 


.622 


.614 


.610 


.605 


.602 


.601 


2. 

2.5 


•637 
.634 


.619 
.617 


.612 
.6lO 


.60S 
.607 


605 
.605 


.604 


.602 
.602 


.604 


3- 


.632 


.616 


.609 


.607 


.605 


.604 


.603 


4. 


.628 


.614 


.608 


.606 


.605 


•603 


.602 


6. 


.623 


.612 


.607 


.605 


.604 


•603 


.602 


S. 


.619 


.610 


.606 


.605 


.604 


• 603 


. C02 


10. 


.616 


.608 


.605 


.604 


.603 


.602 


.601 


20 


.606 


.604 


.602 


.602 


.602 


.601 


.600 


50. 


.602 


.601 


.601 


.600 


.600 


•599 


599 


100. 


•599 


.598 


-593 


.598 


.593 


.598 


•593 



The same general laws of variation are here observed as for 
circular orifices, the coefficient decreasing as the head increases 
and as the size of the square increases. It should be noticed 
that the coefficients are always slightly larger than those for 
circles of the same diameter ; this is perhaps caused by the 
less perfect contraction of the jet due to the corners of the 
square. 

The horizontal lines drawn in the last three columns of the 
table indicate the limit h == 4^; so that the exact formula is to 



82 FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

be used for cases that fall above these lines. The error in the 
use of the approximate formula when k= 3.5^ is about one 
tenth of one per cent, which is probably less than the error in 
applying the coefficient to any given orifice in practice. For 
all values except those above the horizontal lines the error of 
the approximate formula is much less than one-tenth of one 
per cent. 

There are few recorded experiments on large square orifices. 
ELLIS measured the discharge from a vertical orifice 2 feet 
square in an iron plate which furnishes the following results : * 

For h — 2.07 feet, c = 0.61 1 ; 
For h = 3.05 feet, c = 0.597; 
For h = 3.54 feet, c = 0.604; 

which indicate that a mean value of about 0.6 for c is all that 
can be safely stated for large orifices. 

Prob. 54. Find from the table the coefficient of discharge 
for a square whose side is 3 inches when the head on its centre 
is 1.8 feet. 

Prob. 55. Compute the probable actual discharge from a 
vertical orifice one foot square when the head on its upper edge 
is one foot. 

Article 40. Rectangular Vertical Orifices. 

Rectangular vertical orifices with the longest edge hori- 
zontal are frequently employed for the measurement of water. 
If b be the breadth, d the depth, 1i x , h„ , and // the head on the 
upper edge, lower edge, and centre, and c the coefficient of dis- 
charge, the discharge per second is 

q = c.\b\fcg{h}-h}\ (26) 



* Transactions American Society Civil Engineers. 1S76, vol. v. p. 9: 



Art. 40.] RECTANGULAR VERTICAL ORIFICES. 83 

or more simply, if h be greater than 4^, 

q = c . bd Vlgk (26') 

The following values of the coefficient c have been compiled 
and computed from the discussion given by FANNING. 45 * Those 
above the horizontal lines are to be used in the exact formula, 
and those below in the approximate formula. 

TABLE VIII. COEFFICIENTS FOR RECTANGULAR ORIFICES 
1 FOOT WIDE. 



Head 

h 
in Feet. 






Depth c 


)f Orifice i 


n Feet. 




! 


0.125 


0.25 


0.50 


o.75 


1.0 


i-5 


2.0 


0.4 
0.6 


0.634 


0.633 
•633 


O.622 

.619 


O.614 








.633 


0.3 


•633 


•633 


.618 


.612 


O.60S 






I. 


.632 


• 632 


.618 


.612 


.606 


O.626 




i-5 
2. 


.630 
.629 


.631 
.630 


.618 


.611 
.611 


.605 
.605 


.626 
.624 


0.628 
.630 


.617 


2-5 
3- 


.628 
.627 


.628 
.627 


.616 
.615 


.611 


.605 
.605 


.616 
.614 


.627 
.619 


.610 


4- 

6. 

8. 


.624 
.615 
.609 


.624 
.615 
.607 


.614 
.609 
.603 


.609 
.604 
.602 


.605 
.602 
.601 


.612 


.616 
.610 


.606 
.602 


.604 


10. 


.606 


.603 


.601 


.601 


.601 


.601 


.602 


20. 








.601 


.601 


.601 


.602 

— . — . — 



This table shows that the variation of c with the head fol- 
lows the same law as for circles and squares. It is also seen 
that for a rectangle of constant breadth the coefficient of dis- 
charge increases as its depth decreases, from which it is to be 
inferred that for a rectangle of constant depth 'the coefficient 
increases with the breadth, and this is confirmed by other ex- 
periments. The value of c for a rectangular orifice is seen to 



* Treatise on Water Supply Engineering, p. 205. 



84 FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

be but slightly larger than for a square whose side is equal to 
the depth of the rectangle. In selecting a coefficient for use 
with an orifice whose size falls outside the limits of the table r 
it should be borne in mind that large orifices have a smaller 
value of c than small orifices. 

A comparison of the values of c for the orifice one foot square 

with those in the last article shows that the two sets of co- 
efficients disagree, these being about one per cent greater than 
those. This is probably due to the less precise character and 
smaller number of experiments from which they were deduced. 
Further experimental data on rectangular orifices are needed. 

Prob. 56. What head is required to discharge 5 cubic feet 
per second through an orifice 3 inches deep and 12 inches 
long ? 

Prob. 57. What is a probable coefficient of discharge for an 
orifice 3 inches deep and 6 inches long, the head on the upper 
edge being 6 inches ? 

Article 41. The [Mixer's Inch. 

The miner's inch may be roughly defined to be the quantity 

of water which will flow from a vertical standard orifice one 

inch square, when the head on the centre of the orifice is 6J 

inches. From Table VII the coefficient of discharge is seen to 

be about 0.623, and accordingly the actual discharge in cubic 

feet per second is 

0.623 X 8.02 f(xl 

q — — a / — - = O.0255, 

1 144 V 12 "' 

and the discharge in one minute is 

60 X 0.0255 = 1.53 cubic feet. 

The mean value of one miner's inch is therefore about 1.5 cubic 
feet per minute. 

The actual value of the miner's inch, however, differs con- 



Art. 41.] THE MINER'S INCH. 85 

siderably in different localities. BOWIE states that in different 
counties of California it ranges from 1.20 to 1.76 cubic feet per 
minute.* The reason for these variations is due to the fact 
that when water is bought for mining or irrigating purposes 
a much larger quantity than one miner's inch is required, and 
hence larger orifices than one square inch are needed. Thus, 
at Smartsville a vertical orifice or module 4 inches deep and 
250 inches long, with a head of 7 inches above the top edge, 
is said to furnish 1000 miner's inches. Again, at Columbia 
Hill, a module 12 inches deep and I2f inches wide, with a head 
of 6 inches above the upper edge, is said to furnish 200 miner's 
inches. In Montana the customary method of measurement 
is through a vertical rectangle, one inch deep, with a head on 
the centre of the orifice of 4 inches, and the number of miner's 
inches is said to be the same as the number of linear inches in 
the rectangle ; thus under the given head an orifice one inch 
deep and 60 inches long would furnish 60 miner's inches. The 
discharge of this is said to be about 1.25 cubic feet per minute, 
or 75 cubic feet per hour. 

A module is an orifice which is used in selling water, and 
which under a constant head is to furnish a given number of 
miner's inches, or a given quantity per second. The sizes and 
proportions of modules vary greatly in different localities, but 
in all cases the important feature to be observed is that the 
head should be maintained nearly constant in order that the 
consumer may receive the amount of water for which he bar^ 
gains and no more. 

The simplest method of maintaining a constant head is by 
placing the module in a chamber which is provided with a gate 
that regulates the entrance of water from the main reservoir or 
canal. This gate is raised or lowered by an inspector once or 
twice a day so as to keep the surface of the water in the cham- 

* Bowie, Treatise on Hydraulic Mining, p. 268. 



86 FLOW OF WATER THROUGH ORIFICES. [Chap. Iv 

ber at a given mark. This plan though simple is costly, except 
in works where many modules are used, and where a daily in- 
spection is necessary in any event, and it is not well adapted 
to cases where there are frequent and considerable fluctuations 
in the surface of the water in the feeding canal. 

Numerous methods have been devised to secure a constant 
head by automatic appliances ; for instance, the gate which 
admits water into the chamber may be made to rise and fall 
by means of a float upon the surface ; the module itself may 
be made to decrease in size when the water rises, and to in- 
crease when it falls, by a gate or by a tapering plug which 
moves in and out and whose motion is controlled by a float. 
These self-acting contrivances, however, are liable to get out 
of order, and require to be inspected more or less frequently.* 

The use of the miner's inch, or of a module, as a standard 
for selling water, may be said to have a certain advantage in 
simplicity, as it depends merely upon an arbitrary definition. 
It is, however, greatly to be desired for the sake of uniformity^ 
that water should be bought and sold by the cubic foot. Only 
in this way can comparisons readily be made, and the con- 
sumer be sure of obtaining exact value for his money. 

Prob. 58. If a miner's inch be 1.57 cubic feet per minute, 
how many miner's inches will be furnished by a module 2 
inches deep and 50 inches long with a head of 6 inches above 
the upper edge?f 

Article 42. Submerged Orifices. 

It is shown in Art. 26 that the effective head which causes 
the flow from a submerged orifice is the difference in level 
between the two water surfaces. The discharge from such an 

* A cheap and simple method of maintaining a nearly constant head by means 
of an excess weir is described by Foote in the Transactions American Society 
of Civil Engineers for March. 1SS7. 

X See Bowie's Hydraulic Mining, page 125. 



Art. 42.] 



SUBMERGED ORIFICES. 



87 



orifice, its inner edge being a sharp definite corner as in Fig. 
25, has been found by experiment to be somewhat less than 
when the flow occurs freely, or, in other words, the values of 
the coefficients of discharge are smaller than those given in 
the preceding articles. The difference, however, is very slight 
for large orifices and large heads, and for orifices one inch 
square under six inches head is about 2 per cent. 

The following table gives values of the coefficient of dis- 
charge for submerged orifices as determined by the experi- 
ments of Hamilton Smith, Jr. The height of the water on 
the exterior of the orifices varied from 0.57 to 0.73 feet above 
their centres. 

TABLE IX. COEFFICIENTS FOR SUBMERGED ORIFICES 



Effective 
Head in Feet. 




Size of Orifice in 


Feet. 




Circle 

0.05 


Square 
0.05 


Circle 
0.1 


Square 

O.I 


Rectangle 
0.05 X 0.3 


0.5 


O.615 


O.619 


O 603 


O.608 


O 623 


J.O 


.6lO 


.614 


602 


.606 


-622 


i-5 


607 


6l2 


600 


.605 


621 


2.0 


605 


.610 


599 


.604 


. 620 


2.5 


.603 


.608 


598 


.604 


. 619 


3.0 


.602 


.607 


.598 


.604 


.618 


4.0 


.601 


.606 


.598 


.604 





The theoretic discharge from a submerged orifice is the 
same for the same effective head whatever be its distance be- 
low the lower water level. It is not likely, however, that the 
same coefficients of discharge would be found for deeply sub- 
merged orifices as for those submerged but slightly. Experi- 
ments in this direction from which to draw conclusions are 
lacking. 

Prob. 59. An orifice one inch square in a gate such as shown 
in Fig. 7, Art. 14, is 3 feet below the higher water level and 2 



L 




88 FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

feet below the lower water level. Compute the discharge in 
cubic feet per minute. 

Article 43. Suppression of the Contraction. 

When a vertical orifice has its lower edge at the bottom of 
the reservoir, as shown at A in Fig. 26, the particles of water 
flowing through its lower portion move in 
lines nearly perpendicular to the plane of 
the orifice, or the contraction of the jet 
does not form on the lower side. This 
is called a case of suppressed or incom- 
plete contraction. The same thing occurs, 
but in a lesser degree, when the lower edge 
FlG- 26, of the orifice is near the bottom as shown 

at B. In like manner, if an orifice be placed so that one of its 
vertical edges is at or near a side of the reservoir, as at C, 
the contraction of the jet is suppressed upon one side, and if 
it be placed at the lower corner of the reservoir, suppression 
occurs both upon one side and the lower part of the jet. 

The effect of suppressing the contraction is, of course, to 
increase the cross-section of the jet at the place where full con- 
traction would otherwise occur, and it is found by experiment 
that the discharge is likewise increased. Experiments also 
show that more or less suppression of the contraction will 
occur unless each edge of the orifice is at a distance at least 
equal to three times its least diameter from the sides or bottom 
of the reservoir. 

The experiments of Lesbros and BlDONE furnish the 
means of estimating the increased discharge caused by sup- 
pression of the contraction. They indicate that for square 
orifices with contraction suppressed on one side the coefficient 
of discharge is increased about 3.5 per cent, and with contrac- 
tion suppressed on two sides about 7.5 per cent. For a rect- 



Art. 44.] 



ORIFICES WITH ROUNDED EDGES. 



89 



angular orifice with the contraction suppressed on the bottom 
edge the percentages are larger, being about 6 or 7 per cent 
when the length of the rectangle is four times its height, and 
from 8 to 12 per cent when the length is twenty times the 
height. The percentage of increase, moreover, varies with 
the head, the lowest heads giving the lowest percentages. 

It is apparent that suppression of the contraction should 
be avoided if accurate results are desired. The experiments 
from which the above conclusions are deduced were made upon 
small orifices with heads less than 6 feet, and it is not known 
how they will apply to large orifices under high heads. 

Prob. 60. Compute the probable discharge from a vertical 
orifice one foot square when the head on its upper edge is one 
foot, the contraction being suppressed on the lower edge. 

Article 44. Orifices with Rounded Edges. 

If the inner edge of the orifice be rounded, as shown in Fig. 
27, the contraction of the jet is modified, and the discharge is 
increased. With a slight degree of 
rounding, as at A, a partial contrac- 
tion occurs ; but with a more com- 
plete rounding, as at C, the parti- 
cles of water issue perpendicular to 
the plane of the orifice and there is 
no contraction of the jet. If a be 
the area of the least cross-section of 
the orifice, and a that of the jet, the coefficient of contraction 
(Art. 34) is 

c>= a -. 
a 

For a standard square-edged orifice (Fig. 25) the mean value of 
c' is 0.62, but with a rounded orifice c' may have any value be^ 
tween 0.62 and 1.0, depending upon the degree of rounding. 




I 



// 



: 



Fig. 27. 



90 FLOW OF WATER THROUGH ORIFICES. [Chap. IV". 

The coefficient of discharge for square-edged orifices has a 
mean value of about 0.61 ; this is increased with rounded edges 
and may have any value between 0.61 and 1.0, although it is 
not probable that values greater than 0.95 can be obtained 
except by the most careful adjustment of the rounded edges to 
the exact curve of a completely contracted jet. 

A rounded interior edge in an orifice is therefore always a 
source of error when the object of the orifice is the measure- 
ment of the discharge. If a contract provides that water shall 
be gauged by standard orifices, care should always be taken 
that the interior edges do not become rounded either by acci- 
dent or by design. 

Prob. 61. If an orifice with rounded edges has a coefficient of 
contraction of* 0.85 and a coefficient of discharge of 0.75, find 
the coefficient ot velocity. 

Article 45. The Measurement of Water by Orifices. 

In order that water may be accurately measured by the use 
of orifices many precautions must be taken, some of which 
have already been noted, but may here be briefly recapitulated. 
The area of the orifice should be small compared with the size 
of the reservoir in order that velocity of approach may not 
affect the flow (Art. 25). The inner edge of the orifice must 
have a definite right-angled corner, and its dimensions are to 
be accurately determined. If the orifice be in wood, care should 
be taken that the inner surface be smooth, and that it be kept 
free from the slime which often accompanies the flow of water 
even when apparently clear. That no suppression of the con- 
traction may occur, the edges of the orifice should not be nearer 
than three times its least dimension to a side of the reservoir. 

Orifices under very low heads should be avoided, because 
slight variations in the head produce relatively large errors, 
and also because the coefficients of discharge vary more rapidly 



Art. 45.] MEASUREMENT OF WATER BY ORIFICES. 9 1 

and are probably not so well determined as for cases where the 
head is greater than four times the depth. For similar reasons 
very small orifices are not desirable. If the head be very low 
on an orifice, vortices will form which render any estimation of 
the discharge unreliable. 

The measurement of the head, if required with precision, 
must be made with the hook gauge which is described in Art. 
50. For heads greater than two or three feet the readings of 
an ordinary glass gauge placed upon the outside of the reser- 
voir will usually prove sufficient, as this can be read to hun- 
dredths of a foot with accuracy. An error of 0.01 feet when the 
head is 3.00 feet produces an error in the computed discharge 
of less than two-tenths of one per cent ; for, the discharges be- 
ing proportional to the square roots of the heads, ^/T^oi divided 
by 4/S.00 equals 1.0017. For the rude measurements in con- 
nection with the miner's inch a common foot-rule will probably 
suffice. 

The effect of temperature upon the discharge remains to be 
noticed ; this is only appreciable with small orifices and under 
low heads. UNWIN found that the discharge was diminished 
one per cent by a rise of 144 degrees in temperature ; his orifice 
was a circle 0.033 f eet in diameter under heads ranging from 
1.0 to 1.5 feet. SMITH found that the discharge was dimin- 
ished one per cent by a rise of 55 degrees in temperature ; his 
orifice was a circle 0.02 feet in diameter, under heads ranging 
from 0.56 to 3.2 feet. This is a further reason why small ori- 
fices and low heads are not desirable in precise measurements 
of discharge. 

The coefficients given. in the preceding tables may be sup- 
posed liable to a probable error of two or three units in the 
third decimal place; thus a coefficient 0.615 should really be 
written 0.615 ± 0.003 ; that is, the actual value is as likely to 
be between 0.612 and 0.618 as to be outside of those limits. 



Cp FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

The probable error in computed discharges due to the coeffi- 
cient is hence about one-half of one per cent. To this are 
added the errors due to inaccuracy of observation, so that it is 
thought that the probable error of careful work with standard 
circular orifices is at least one per cent. The computed dis- 
charges are hence liable to error in the third significant figure, so 
that it is useless to carry numerical results beyond four figures 
when based upon tabular coefficients. As a precise method of 
measuring small quantities of water, standard orifices take a 
high rank when the observations are conducted with care. 
With rectangular orifices the probable error is liable to be two 
per cent or more. 

Prob. 62. What error is produced in the computed discharge 
if the head be read 1.38 feet when it should have been 1.385 
feet? 

Article 46. The Energy of the Discharge. 

A jet of water flowing from an orifice possesses by virtue 
of its velocity a certain energy or potential work, which is al- 
ways less than the theoretic energy due to the head (Art. 31). 
Let h be the head and W the weight of water discharged per 
second, then the theoretic energy per second is 

K= Wh. 

Let v be the actual velocity of the water at the contracted sec- 
tion of the jet ; then the actual energy per second of the water 
as it passes that section is 

k=W~ (27) 

But — is less than h because v is less than the theoretic ve- 
locity; or, if c l be the coefficient of velocity (Art. 36), 

v = c\ \ 2gh, 



Art. 46.] THE ENERGY OF THE DISCHARGE. 93 

whence — — == c?h ; 

2g 

and hence the effective energy is 

k — c^Wh (27') 

The efficiency of the jet accordingly is 

k 

which is always less than unity. 

For the standard orifice with square inner edges a mean 
value of c l is 0.98. The mean effective energy of the jet at the 
contracted section is hence 

k = 0.96 Wh ; 

that is, the effective energy is 96 per cent of the theoretic. For 
high heads c x is greater than 0.98, and the efficiency becomes 
greater than 96 per cent. It is not possible in practice to take 
advantage of this high efficiency, on account of the difficulty of 
placing the vanes of a hydraulic motor so near the orifice, and 
accordingly standard orifices are never used when the work of 
the discharge is to be utilized. 

The loss of energy, or potential work, is hence about 4 per 
cent with the standard orifice. This is caused by the influence 
of the edges of the orifice which retard the velocity of the 
outer filaments of the jet, That these outer filaments move 
slower than the central ones may be seen by placing fine sand 
or sawdust in the water and observing that the greater part 
passes out of the orifice in the interior of the jet. 

Prob. 63. Prove that the energy due to the velocity of the 
jet in the plane of the inner edge of the standard orifice is 
about 37 per cent of the theoretic energy. How is the remain- 
ing 63 per cent accounted for? 



94 FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

Article 47. "Discharge under a Dropping Head. 

If a vessel or reservoir receives no inflow of water while an 
orifice is open, the head drops and the discharge decreases in 
each successive second. Let H be the head on the orifice at a 
certain instant, and h the head t seconds later ; let A be the 
area of the uniform horizontal cross-section of the vessel, and a 
the area of the orifice. Then, as demonstrated in Art. 28, the 
time t is 

•7/4 

t = ^±L(VH- Vh). 
aV2g 

This is the theoretic time ; to determine the actual time the 
coefficient of discharge must be introduced. Referring to the 
demonstration, it is seen that a \ ' 2gy . dt is the theoretic dis- 
charge in the time dt\ hence the actual discharge is c . a \ / 2gy dt, 
and accordingly the above equation is to be thus modified : 

t = ^{VH-Vh), (28) 

ca V2g 

which is the practical formula for the time in which the water 
level drops from H to //. In using this formula c may be taken 
from the tables in the preceding articles, an average value being 
selected corresponding to the average head. 

Experiments have been made to determine the value of c 
by the help of this formula ; the liquid being allowed to flow, 
A, a, If, h, and t being observed, whence c is computed. In 
this way c for mercury has been found to be about 0.62.* Only 
approximate mean values can be found in this manner, since c 
varies with the head, particularly for small orifices (Art. 38). 
For a large orifice the time of descent is usually so small that 
it cannot be noted with precision, and the friction of the liquid 

*Downing's Elements of Practical Hydraulics (Xondon, 1875), P- x -7- 



Art. 47.] DISCHARGE UNDER A DROPPING HEAD. 95 

on the sides of the vessel may also introduce an element of un- 
certainty. This experiment has therefore little value except 
as illustrating and confirming the truth of the theoretic formulas. 

The discharge in one second when the head is H at the 
beginning of the second is found as follows : The above equa- 
tion may be written in the form 

2A 

By squaring both members, transposing and multiplying by A, 
this becomes 

A(H-k) = tca^(vH-'^~^}- 

But the first term of this equation is the quantity discharged 
in t seconds ; therefore the discharge Q for t seconds may be 
written 

and the discharge in one second is 



(•'5*-S-) 



g = ca[V2gH-Cjj\ (29) 

If A = co, this becomes ca V2gH> which should be the case, 
for then H would remain constant. The head at the end of one 

second is// = H j, and at the end of t seconds is// = H — —-.. 

A A 

For example, let an orifice one foot square in a reservoir of 
10 square feet section be under a head of 9 feet, and c — 0.602. 
Then the discharge in one second is 13.9 cubic feet, and the 
head drops to 7.61 feet. The discharge in the next second is 
1 2, 7 cubic feet, and the head drops to 6.34 feet. 



96 FLOW OF WATER THROUGH ORIFICES. [Chap. IV. 

Prob. 64. Find the time required to discharge 480 gallons 
from an orifice 2 inches in diameter at 8 feet below the water 
level in a tank which is 4 X 4 feet in cross-section. 



Article 48. Emptying and Filling a Canal Lock. 

A canal lock is emptied by opening one or more orifices in 
the lower gates. Let a be their area, and H the head of water 
on them when the lock is full ; let A be the area of the hori- 
zontal cross-section of the lock. Then in the formula of the 
last article, h = o, and the time of emptying the lock is 

2AVH 

* = 77= (30; 

ca V2g 

If the discharge be free into the air, H is the distance from the 
centre of the orifice to the level of the water in the lock when 
filled ; but if, as is usually the case, the orifices be below the 
level of the water in the tail bay, H is the difference in height 
between the two water levels. The tail bay is regarded as so 
large compared with the lock that its water level remains con- 
stant. 

For example, let it be required to find the time of empty- 
ing a canal lock 80 feet long and 20 feet wide through two 
orifices, each of 4 square feet area, the head upon which is 16 
feet when the lock is filled. Using for c the value 0.6 for orifices 
with square inner edges, the formula gives 

2 X 80 X 20 X 4 , '. 

*■ = 0.6 X 8 X 8.02 = 333 SeConds = ^ minutes - 

If, however, the circumstances be such that c is 0.8, the time is 
about 250 seconds, or 41 minutes. It is therefore seen that it 
is important to arrange the orifices of discharge in canal locks 
with rounded inner edges. 



Art. 48.] EMPTYING AND FILLING A CANAL LOCK. 



97 



The filling of the lock is the reverse operation. Here the 
water in the head bay remains at a constant level, and the dis- 
charge through the orifices in the upper gates decreases with 
the rising head in the lock. Let H be the effective head on 
the orifices when the lock is empty, and y the effective head 
at any time t after the beginning of the discharge. The area 
of the section of the lock being A, the quantity Ady is dis- 




Head Bay_ZJT ~ ~ 

_ 



Lock 



_ Tail Bay 



& 



Fig. 28. 



charged in the time dt, and this is equal to ca V2gy St, if a be 
the area of the orifices and c the coefficient of discharge. 
Hence the same expression as (30) results, and the times of 
filling and emptying a lock are equal if the orifices are of the 
same dimensions and under the same heads. The area a for 
any case is found from (30), in which A, If, and t are given. 

Prob. 65. Compute the areas of the two orifices when 
A = 1800 square feet, t = 3 minutes, c = 0.7, H = 7 feet for 
the upper and 12 feet for the lower orifice. 



93 



FLOW OF WATER OVER WEIRS. 



[Chap. 



CHAPTER V. 
FLOW OF WATER OVER WEIRS. 

Article 49. Description of a Weir. 

A weir is a notch in the top of the vertical side of a vessel 
or reservoir through which water flows. The notch is generally 
rectangular, and the word weir will be used to designate a rect- 
angular notch unless otherwise specified, the lower edge of the 
rectangle being truly horizontal, and its sides vertical. The 
lower edge of the rectangle is called the " crest" of the weir. 




Fig. 29. 

In Fig. 29 are shown the outlines of two kinds of weirs, A be- 
ing the more usual form where the vertical edges of the notch 
are sufficiently removed from the sides of the reservoir or feed- 
ing canal, so that the sides of the stream may be fully con- 
tracted ; this is called a weir with end contractions. In the form 
at £, the edges of the notch are coincident with the sides of 
the feeding canal, so that the filaments of water along the sides 
pass over without being deflected from the vertical planes in 
which they move ; this is called a weir without end contrac- 
tions, or with end contractions suppressed. 

It is necessary in order to make accurate measurements of 
discharge by a weir that the same precaution should be taken 



Art. 49.] 



DESCRIPTION OF A WEIR. 



99 




Fig. 



as for orifices (Art. 34), namely, that the inner edge of the 
notch shall be a definite angular corner so that the water 
in flowing out may touch the crest only in a line, thus insur- 
ing complete contraction. In precise observations a thin 
metal plate will be used for a crest as 
seen in Fig. 30, while in common work 
it may be sufficient to have the crest 
formed by a plank of smooth hard 
wood with its inner corner cut to a 
sharp right angle and its outer edge 
bevelled. The vertical edges of the weir should be made in 
the same manner for weirs with end contractions, while for 
those without end contractions the sides of the feeding canal 
should be smooth and be prolonged a slight distance beyond 
the crest. It is also necessary to observe the same precautions 
as for orifices to prevent the suppression of the contraction 
(Art. 43), namely, that the distance from the crest of the weir to 
the bottom of the feeding canal, or reservoir, should be greater 
than three times the head of water on the crest. For a weir 
with end contractions a similar distance should exist between 
the vertical edges of the weir and the sides of the feeding canal. 



The head of water H upon the crest of a weir is usually 
much less than the breadth of the crest, b. The value of H 
should not be less than 0.1 foot, and it rarely exceeds 1.5 feet« 
The least value of b in practice is about 0.5 feet, and it does 
not often exceed 20 feet. Weirs are extensively used for 
measuring the discharge of streams, and for determining the 
quantity of water supplied to hydraulic motors ; the practical 
importance of the subject is so great that numerous experi- 
ments have been made to ascertain the laws of flow, and the 
coefficients of discharge. 

Prob. 66. If a feedine canal three feet wide discharees 12 



cubic feet per second when the water 
the mean velocity of flow ? 



is 2 feet deep, what is 



IOO 



FLOW OF WATER OVER WEIRS. 



[Chap. V, 




Fig. ax. 



Article 50. The Hook Gauge. 

As the head on the crest of a weir is low it 
must be determined with precision in order to 
avoid error in the computed discharge (Art. 45). 
The hook gauge, invented by Boyden about 
1840, consists of a rod sliding vertically in fixed 
supports, the amount of vertical motion being 
determined by the readings of a vernier. The 
vernier can be set to read 0.000 when the sharp 
point of the hook is on the same level as the 
crest of the weir ; when the water is flowing 
over the crest, the rod is raised by the slow- 
motion screw until the point of the hook is at 
the water level. Before the point pierces the 
surface or skin of the water, a pimple or pro- 
tuberance is seen to rise above it due to capil- 
lary action ; the hook is then depressed until 
this pimple is barely perceptible, when the point 
is at the true water level. The head of water 
on the crest is then indicated by the reading 
of the scale and vernier. The best hook gauges 
are made to read to ten-thousandths of a foot, 
and it has been stated that an experienced ob- 
server can in a favorable light detect differences 
in level as small as 0.0002 feet. The surface 
of water at the hook must be perfectly quiet, 
and hence a box without a bottom or with 
openings to admit the water is often placed 
around it. Fig. 31 shows the hook gauge as 
arranged by EMERSON.* 



* Emerson's Hydrodynamics (Springfield. Mass.. iSSt\ p. 56. 



Art. 50.] THE HOOK GAUGE. IOI 

A cheaper form of hook gauge, and one sufficiently precise 
in some classes of work, can be made by screwing a hook into 
the foot of a levelling rod. The back part of the rod is then 
held in a vertical position by two clamps on fixed supports, 
while the front part is free to slide. It is easy to arrange a 
slow-motion movement so that the point of the hook may be pre- 
cisely placed at the water level. The reading of the vernier is 
determined when the point of the hook is on the same level as 
the crest of the weir, and by subtracting from this the subse- 
quent readings the heads of water are known. A New York 
levelling rod reading to thousandths of a foot is to be preferred. 

The greatest error of a hook gauge is thought to be in set- 
ting it for the level of the crest. In the larger forms of hooks 
this may be done by taking elevations of the crest, and of the 
point of the hook by means of an engineers level and a light 
rod. With smaller hooks it may be done by having a stiff 
permanent hook the elevation of whose point with respect to 
the crest is determined by precise levels ; the water is then al- 
lowed to rise slowly until it reaches the point of this stiff hook, 
when readings of the vernier of the lighter hook are taken. 
Another method is to allow a small depth of water to flow over 
the crest and to take readings of the hook, while at the same 
time the depth on the crest is measured by a finely graduated 
scale. Still another way is to allow the water to rise slowly, 
and to set the hook at the water level when the first filaments 
pass over the crest ; this method is not a very precise one on 
account of capillary attraction along the crest. As the error 
in setting the hook is a constant one which affects all the sub- 
sequent observations, especial care should be taken to reduce 
it to a minimum by taking a number of observations from 
which to obtain a precise mean result. 

In rough gaugings of streams the precision of a hook gauge 
is often not required, and the heads may be determined by 



102 FLOW OF WATER OVER WEIRS. [Chap. V. 

simpler methods. For example, a post may be set with its top 
on the same level as the crest of the weir, and the depth of 
water over the top of the post be measured by a scale gradu- 
ated to tenths and hundredths of a foot, the thousandths be- 
ing either estimated or omitted entirely. 

The head H on the crest of the weir is in all cases to be 
measured several feet up stream from the crest, as indicated in 
Fig. 30. This is necessary because of the curve taken by the 
surface of the water in approaching the weir. The distance to 
which this curve extends back from the weir depends upon 
many circumstances (Art. 59), but it is considered that perfect- 
ly level water will be found at 2 or 3 feet distance back for 
small weirs, and at 6 or 8 feet for very large weirs. It is de- 
sirable that the hook should be placed at least one foot from 
the sides of the feeding canal, if possible. As this is apt to 
render the position of the observer uncomfortable, some ex- 
perimenters have placed the hook in a pail at a few feet dis- 
tance from the canal, the water being led to the pail by a pipe : 
this pipe should enter the feeding canal several feet above the 
crest, and the water should enter it, not at its end, but through 
a number of holes drilled at intervals along its circumference. 

Prob. 67. Show by using formula (9)' of Art. 22 that an 
error of about one-half of one per cent results in the discharge 
if an error of 0.001 feet be made in reading the head when 
H — 0.3 feet. 

Article 51. Formulas for the Discharge. 

The theoretic discharge through a rectangular notch or 
weir was found in Art. 22 to be 

Q = f V^. bH* % 

— * 00 

in which b is the breadth of the notch, commonly called the 
length of the weir, and H the depth of water on the lower 



Art. 51.] FORMULAS FOR THE DISCHARGE. 103 

edge. It might be inferred that this depth is that in the plane 
of the weir; but as the deduction of the formula supposes 
nothing regarding the fall due to the surface curve, and regards 
the velocity at any point above the crest as due to the head 
upon that point below the free water surface, it seems that H 
should be measured with reference to that surface, as is actu- 
ally done by the hook gauge. The above formula then gives 
the theoretic discharge per second, provided that there be no 
velocity at the point where H is measured, which can only be 
the case when the area of the weir opening is very small com- 
pared to that of the cross-section of the feeding canal. This 
condition would be fulfilled for a rectangular notch placed at 
the side of a large pond. 

When there is an appreciable velocity of approach of the 
water at the point where H is measured by the hook gauge, 
the above formula must be modified. Let v be the mean 
velocity in the feeding canal at this section ; this velocity may 
be regarded as due to a fall, k, from the surface of still water 

at some distance up stream from - 

the hook, as shown in Fig. 32. -_=_ — ^— :ps ^^ 

Now the true head on the crest of _~~ ==!p=£^?^3|^ 

the weir is H -\- h, as this would =&. — r — — q=I \\m 

have been the reading of the - w/////* <s/,«/<„ws7z?%7/sb 
hook gauge had it been placed FlG - 32. 

where the water had no velocity. Accordingly the theoretic 
discharge is 

Q = l\^g.b(H + h)i, 

in which H is read by the hook and h is determined from the 
mean velocity v. 

The actual discharge per second is always less than the 
theoretic discharge, due to the contraction of the stream and 
the resistances of the edges of the weir. To take account of 



104 FLOW OF WATER OVER WEIRS. [Chap. V. 

these a coefficient is applied to the theoretic formulas in the 
same manner as for orifices ; these coefficients being deter- 
mined by experiment, the formulas may then be used for 
computing the actual discharge. It has also been proposed by 
Smith to modify the velocity-head h, owing to the fact that 
the velocity of approach is not constant throughout the 
section, but greater near the surface than near the bottom, as 
in streams (Art. 107). Accordingly the following may be 
written as an expression for the actual discharge : 

q = c.\^.b{H-\-nKf (31) 

in which c is the coefficient of discharge whose value is always 
less than unity, and n is a number which lies between 1.0 
and 1.5.* 

The above formulas are not in all respects perfectly satis- 
factory, and indeed many others have been proposed. The 
actual discharge differs, however, so much from the theoreti- 
cal that the final dependence must be upon the coefficients 
deduced from experiment, and hence any fairly reasonable 
formula may be used within the limits for which its coefficients 
have been established. In spite of the objections which may 
be raised against all forms of formulas, the fact remains that 
the measurement of w r ater by weirs is one of the most con- 
venient methods, and probably the most precise method, unless 
the quantity is so small as to pass through a circular orifice 
less than one foot in diameter. With proper precautions the 
probable error in measurements of discharge by weirs should 
be less than two or three per cent. 

Prob. 68. Find the velocity-head h when the mean velocity 
of approach is 20 feet per minute. 

* Smith's Hydraulics, p. S3. 



Art. 52.] VELOCITY OF APPROACH. I05 

Article 52. Velocity of Approach. 

The velocity-head h, which produces the mean velocity of 
approach v is (Art. 20) 

h=—z=: 0.01555^. 

2 g 

Accordingly to obtain h the value of v must be determined. 
One way of doing this is to observe the time of passage of a 
Hoat through a given distance; but this is not a precise method. 
The usual method is to compute v from an approximate value 
of the discharge, which is first computed by regarding v, and 
hence h, as zero. This determination is rendered possible by 
the fact that p is usually small, and hence that h is quite small 
as compared with H. 

Let B be the breadth of the cross-section of the feeding 
canal at the place where the readings of the hook are taken, 
and let G be its depth below the crest (Fig. 32). The area of 
that cross-section then is 

A = B(G + H). 

The mean velocity in this section now is 

in which q' is found from the formula 

q' = c\tfzg. bHK 

This value of q' is an approximation to the actual discharge ; 
from it v is found, and then h, after which the discharge q can 
be computed. If thought necessary, h may be recomputed by 
using q instead of q' ; but this will rarely be necessary. 

For example, the small weir with end contractions used in 
the hydraulic laboratory of Lehigh University has B = 7.82 



106 FLOW OF WATER OVER WEIRS. [Chap. V. 

feet and G = 2.5 feet. The length of the weir b is adjustable 
according to the quantity of water delivered by the stream. 
On April 10, 1888, the value of b was 1.330 feet, and values of 
i7 ranged from 0.429 to 0.388 feet. It is required to find the 
velocity v and the velocity-head h, when H = 0.429 feet. Here 
the coefficient c is 0.602 (Art. 53), hence the approximate dis- 
charge per second is 

q' = 0.602 X I X 8.02 X 1-33 X 0.429!, 

or q' = 1.203 cubic feet per second. 

The mean velocity of approach then is 

I - 2 °3 

v — 7 ; — = 0.053 feet per second, 

(2.5 + 0.4J7.82 DJ r ' 

from which the velocity-head h is 

0-°53 2 
h — - = 0.00004 feet. 
64.32 

This is too small to be regarded, since the hook gauge used 
determines the heads only to thousandths of a foot. 

The velocity-head h may be directly expressed in terms of 
the discharge by substituting for v its value —7 ; thus : 

A = o.oiS5s(-|) (32) 

In general, this expression will be found the most convenient 
one for computing the value of the head corresponding to the 
velocity of approach. 

With a weir opening of given size under a given head H, 
the velocity of approach is less the greater the area of the sec- 
tion of the feeding canal, and it is desirable in building a weir 
to make this area large so that the velocity v may be small. 



Art. 53.] WEIRS WITH END CONTRACTIONS. 107 

For large weirs, and particularly for those without end con- 
tractions, v is sometimes as large as one foot per second, giving 
h =0.0155 feet, and these should be regarded as the highest 
values allowable if precision of measurement is required. 

Prob. 69. FTELEY and Stearns' large suppressed weir had 
the following dimensions : b > = B = 18.996 feet, G— 6.55 feet, 
and the greatest measured head was 1.6038 feet. Taking 
c = 0.622, compute the velocity of approach and its velocity- 
head. 

Article 53. Weirs with End Contractions. 
Let b be the breadth of the notch or length of the weir, H 
the head above the crest measured by the hook gauge, and c 
an experimental coefficient. Then if there be no velocity of 
approach the discharge per second is 

q = c.\v^.bm (33) 

But if the mean velocity of approach at the section where the 
hook is placed be v, let h be the head which would produce 
this velocity. Then the discharge per second is 

q = c.%V r ^.b(H+i.4kf (33)' 

The quantity H -\- 1.4^ is called the effective head on the crest, 
and, as shown in the last article, h is usually small compared 
with H. 

The following table contains values of the coefficient of 
discharge c as deduced by HAMILTON Smith, Jr.,* from a 
discussion of the experiments made by LESBROS, Francis, 
Fteley and STEARNS, and others. In these experiments q is 
determined by actual measurement in a vessel of large size, and 
the other quantities being observed c is computed. Values of 
c for different lengths of weir and for different heads are thus 

* Hydraulics (London and New York, 1884), p. 132. 



icS 



FLOW OF WATER OVER WEIRS. 



[Chap. V, 



obtained, which being plotted enable mean curves to be drawn, 
from which intermediate values are taken. The heads in the 
first column are the effective heads H-\- 1.4/1 ; but as h is small, 
little error can result in using H as the argument with which to 
enter the table in selecting a coefficient. 



TABLE X. COEFFICIENTS FOR CONTRACTED WEIRS. 



Effective 

Head 
in Feet. 


Length of Weir in Feet. 


0.66 


1 


2 


3 


5 


10 


19 


1 

0.1 


0.632 


639 


646 


0.652 


O.653 


0.655 


0.656 


O.15 


.619 


625 . 


634 


.638 


.640 


.641 


.642 


0.2 


.611 


618 


626 


.630 


.631 


.633 


.634 


O.25 


.605 


612 


621 


.624 


.626 


.623 


.629 


0.3 


.601 


6o3 


616 


.619 


.621 


.624 


.625 


0.4 


•595 


601 


609 


.613 


.615 


.6i3 


.620 


0-5 


•59° 


596 


605 


.608 


.611 


.615 


.617 


Os6 


.5S7 


593 


601 


.605 


.60S 


.613 


.615 


0.7 




590 


59S 


• 603 


.606 


.612 


.614 


o.S 






595 


.600 


.604 


.611 


.613 


0.9 






592 


.593 


.603 


.609 


.612 


1.0 






59° 


•595 


.601 


.6o3 


.611 


1.2 






5S5 


•59i 


• 597 


.605 


.610 


1.4 






5 So 


0S7 


•594 


.602 


.609 


1.6 








.5S2 


091 


.600 


.607 



It is seen from the table that the coefficient increases with 
the length of the weir, which is due to the influence of the end 
contractions being independent of the length. The coefficient 
also increases as the head on the crest diminishes. The table 
also shows that the greatest variation in the coefficients occurs 
under small heads, which are hence to be avoided in order to 
secure accurate measurements of discharge. 

Interpolation may be made in this table for heads and 
lengths of weirs intermediate between the values given, regard- 



Art. 53.] WEIRS WITH END CONTRACTIONS. 1 09 

ing the coefficients as varying uniformly ; but it will be better 
in any actual case to diagram the coefficients on cross-section 
paper, from which the interpolation can be made more easily 
and accurately. 

As an example of the use of the formula and table, let it be 
required to find the discharge per second over a weir 4 feet 
long when the head H is 0.457 f eet > there being no velocity of 
approach. From the table the coefficient of discharge is 0.614 
for i7=o.4 an d 0.6095 for //=o.5, which gives about 0.612 
when H •=■ 0.457. Then the discharge per second is 

q — 0.612 X I X 8.02 X 4 X 0.4575 = 4.04 cubic feet 

If the width of the feeding canal be 7 feet, and its depth 
below the crest be 1.5 feet, the velocity-head is 

h = o.oi555^-4:^--) == 0.00134 feet. 

The effective head now becomes 

H -\- 1.4/1 — 0.459 f eet > 

and the discharge per second is 

q — 0.612 X I X 8.02 X 4 X 0.459^ = 4-°7 cubic feet. 

It is to be observed that the reliability of these computed dis- 
charges depends upon the precision of the observed quanti- 
ties and upon the coefficient c\ this is probably liable to an 
error of one or two units in the third decimal place, which is 
equivalent to a probable error of about three-tenths of one per 
cent. On the whole, regarding the inaccuracies of observation, 
a probable error of one per cent should at least be inferred, so 
that the value q = 4.07 cubic feet per second should strictly be 
written, 

q = 4-07 ± 0.04 ; 



HO FLOW OF WATER OVER WEIRS. [Chap. V. 

that is to say, the discharge per second has 4.07 cubic feet for 
its most probable value, and it is as likely to be between the 
values 4.03 and 4. 11 as to be outside of those limits. 

Prob. 70. Compute the discharges per second through a 
weir whose length is 2.5 feet, width of feeding canal 6 feet, 
depth below crest 1.6 feet when the heads on the crest are 
0.314, 0.315, and 0.316 feet. 

Prob. 71. Compute the coefficient of discharge for the fol- 
lowing experiment by FRANCIS, in which q was found by actual 
measurement in a large tank: £ = 9.997 feet, B = 13.96 feet, 
G =4.19 feet, 77=1.5243 feet, 2^=64.3236 and ^ = 61.282 
cubic feet per second. Ans. c = 0.602. 

Article 54. Weirs without End Contractions. 

For weirs without end contractions, or suppressed weirs, 
when there is no velocity of approach, the discharge per second 
is 

q = *.\V5g.bH*; ,.-.. (34) 

and when there is velocity of approach, 

q=c\lft£. b{H-\-\\Kf. . . . (34)' 

Here the notation is the same as in the last article, and c is to 
be taken from the following table, which gives the coefficients 
of discharge as deduced by SMITH. 

It is seen that the coefficients for suppressed weirs are 
greater than for those with end contractions : this of course 
should be the case, as contractions diminish the discharge. 
They decrease with the length of the weir, while those for 
contracted weirs increase with the length. Their greatest 
variation occurs under low heads, where they rapidly increase 
as the head diminishes. It should be observed that these 
coefficients are not reliable for lengths of weirs under 4 feet, 



Art. 54.] WE IKS WITHOUT END CONTRACTIONS. 



II I 



owing to the few experiments which have been made for 
short weirs. Hence, for small quantities of water, weirs with 

TABLE XI. COEFFICIENTS FOR SUPPRESSED WEIRS. 



Effective 
Head 

in 
Feet. 






Length c 


f Weir in 


Feet. 






19 


10 


7 


5 


4 


3 


2 


O.I 


0.657 


0.658 


O.658 


O.659 








O.15 


.643 


.644 


.645 


.645 


O.647 


O.649 


O.652 


0.2 


.635 


•637 


.637 


• 6 3 S 


.641 


.642 


.645 


O.25 


.630 


.632 


.633 


.634 


.636 


.638 


.641 


0.3 


.626 


.628 


.629 


.631 


.633 


.636 


•639 


0.4 


.621 


.623 


.625 


.628 


.630 


.633 


.636 


0.5 


.619 


.621 


.624 


.627 


.630 


.633 


•637 


0.6 


.618 


.620 


,623 


.627 


.630 


•634 


.638 


0.7 


.618 


.620 


.624 


.628 


.631 


.635 


.640 


0.8 


.618 


.621 


• 625 


.629 


•633 


.637 


.643 


0.9 


.619 


.622 


.627 


.631 


.635 


.639 


•645 


1.0 


.619 


.624 


.628 


.633 


.637 


.641 


.648 


1.2 


.620 


.626 


.632 


.636 


.641 


.646 




1.4 


.622 


.629 


.634 


.640 


.644 






1.6 


.623 


.631 


•637 


.642 


.647 







end contractions should be built in preference to suppressed 
weirs. For a weir of infinite length it would be immaterial 
whether end contractions existed or not ; hence for such a 
case the coefficients lie between the values for the "19-foot 
weir in Table X. and those for the 19-foot weir in the table 
here given. 

For a numerical illustration the same data as in the ex- 
ample of the last article will be used, namely, b = 4 feet, 
G = 1.5 feet, and H = 0.457 feet. The coefficient from the 



112 FLOW OF WATER OVER WEIRS. [Chap. V. 

table is 0.630 ; then for no velocity of approach the discharge 
per second is 

q = 0.630 X f X 8.02 X 4 X 0.4S7 1 = 4- 16 cubic feet. 

Here the width B would probably be also 4 feet ; the head 
corresponding to the velocity of approach then is 

h = 0.01555^-j^gJ = 0.0044 feet, 

and the effective head is 

H -\- \\h — 0.463 feet, 

from which the discharge per second is 

q — 0.630 X f X 8.02 X 4 X 0.463^ =± 4.24 cubic feet. 

This shows that the velocity of approach exerts a greater in- 
fluence upon the discharge than in the case of a weir with end 
contractions. 

Prob. 72. Compute the discharge per second over a weir 
without end contractions when b = 9.995 feet, H — 0.7955 
feet, G = 4.6 feet. Ans. q = 23.7 cubic feet per second. 

Article 55. Francis' Formulas. 

The formulas most extensively used for computing the 
flow through weirs are those established by FRANCIS in 1854* 
from the discussion of his numerous and carefully conducted 
experiments, but as they are stated without tabular coeffici- 
ents they are to be regarded as giving only mean approximate 
results. The experiments were made on large weirs, most of 
them 10 feet long, and with heads ranging from 0.4 to 1.6 feet, 
so that the formulas apply particularly to such, rather than to 
short weirs and low heads. The length b and the head H being 

* Lowell Hydraulic Experiments (4th edition, New York, 1883), p. 133. 



Art. 55.] FRANCIS' FORMULAS. 1 13 

expressed in feet, the discharge per second, when there is no 
velocity of approach, is, for weirs without end contractions, or 
suppressed weirs, 

q= 3-33^; (35) 

and for weirs with end contractions, 

q = 3.33 (b-0.2H)Hh ...... (36) 

Here it is regarded that the effect of each end contraction is 
to diminish the effective length of the weir by 0.1H. 

FRANCIS' method of correcting for velocity of approach 
differs from that of SMITH, and is the same as that explained 
in Art. 25. The head h causing the velocity of approach is 
computed in the usual way, and then the formulas are written, 
for weirs without end contractions, 

* = 3-33*[(fl' +*)»-#]; • • • • (35)' 
and for weirs with end contractions, 

<? = 3 . 3i (i-o.2B)[(H + A)i-Al]. . . . (36)' 

It is necessary that this method of introducing the velocity of 
approach should be strictly observed, since the mean number 
3.33 was deduced for this form of expression. 

It is seen that the number 3.33 is c .\ V2g, where c is the 
true coefficient of discharge. The 88 experiments from which 
this mean value was deduced show that the coefficient 3.33 
actually ranged from 3.30 to 3.36, so that by its use an error 
of one per cent in the computed discharge may occur. When 
such an error is of no importance the formula may be safely 
used for weirs longer than 4 feet and heads greater than 0.4 
feet. 

Prob. 73. Find by Francis' formulas the discharge when 
B — 7 feet, b — 4 feet, H — 0.457 f eet > and G = 1.5 feet, the 
weir being one with end contractions. 




114 FLOW OF WATER OVER WEIRS. [Chap. V. 



Article 56. Submerged Weirs. 

When the water on the down-stream side of the weir is al- 
lowed to rise higher than the level of the crest the weir is said 
to be submerged. In such cases an entire change of condition 
results, and the preceding formulas are inapplicable. Let H be 
the head above the crest measured up stream from the weir by 
the hook gauge in the usual manner, and let H' be the head 
above the crest of the water down stream from the weir meas- 
ured by a second hook gauge. If H be constant, the discharge 

is uninfluenced until the lower water 
rises to the level of the crest, provided 
that free access of air is allowed be- 
neath the descending sheet of water. 
But as soon as it rises slightly above 
the crest so that H' has small values, 
the contraction is suppressed and the discharge hence increased. 
As H' increases, however, the discharge diminishes until it be- 
comes zero when H' equals H. Submerged weirs cannot be 
relied upon to give precise measurements of discharge on 
account of the lack of experimental knowledge regarding them, 
and should hence always be avoided if possible. 

The following method for estimating the discharge over 
submerged weirs without end contractions is taken from the 
discussion given by HERSCHEL * of the experiments made by 
FRANCIS and by FTELEY and STEARNS. The observed head H 
is first multiplied by a number n, which depends upon the 
ratio of H' to H, and then the discharge is to be found by the 
formula 

* Transactions American Society of Civil Engineers, 1SS5, vol. xiv. p. 194. 



Art. 56.] 



SUBMERGED WEIRS. 



115 



The values of n are given in the following table 

TABLE XII.; SUBMERGED WEIRS. 



H> 
H 


n 


H 


- 


H 


n 


H 
H 


n 


O.OO 


I. OOO 


O.18 


O.989 


O.38 


0-935 


O.58 


O.856 


.01 


I.OO4 


.20 


O.985 


.40 


O.929 


.60 


O.846 


.02 


I.006 


.. .22 


O.980 


.42 


O.922 


.62 


O.836 


.04 


I.007 


.24 


0-975 


•44 


O.915 


.64 


O.824 


.06 


I.607 


.26 


O.97O 


.46 


O.908 


.66 


O.813 


.08 


1.006 


.28 


O.964 


.48 


O.gOO 


.70 


O.787 


.IO 


1.005 


•30 


o-959 


•50 


O.892 


•75 


0.750 


.12 


I.002 


.32 


o.953 


•52 


O.884 


.80 


O.703 


.14 


0.998 


•34 


o.947 


•54 


O.875 


.90 


0.574 


.16 


O.994 


•36 


0.941 


.56 


O.866 


1. 00 


O.OOO 



The numbers in this table are liable to a probable error of 
about one unit in the second decimal place when H' is less than 
0.2H, and to greater errors in the remainder of the table, those 
values of n less than 0.70 being in particular uncertain. This 
discussion shows that H' may be nearly one-fifth of //without 
affecting the discharge more than two per cent. 

A rational formula for the discharge over submerged weirs 
may be deduced in the following manner. The theoretic dis- 
charge may be regarded as composed of two portions, one 
through the upper part H — H ; , and the other through the 
lower part H' . The portion through the upper part is given 
by the usual weir formula, H — H' being the head, or 

and that through the lower part is given by the formula for a 
submerged orifice (Art. 42), in which b is the breadth, H' the 
height, and H — H' the effective head, or 

& = bH' V2g{H-H'). 



Il6 FLOW OF WATER OVER WEIRS. [Chap. V. 

The addition of these gives the total theoretic discharge, 

This may be put into the more convenient form, 

Q = i *Kgb(H+ iH')(H-Hy. 

The actual discharge per second may now be written, 

qi = c.l^b{H + kH\H-HJ;. . (37) 

in which c is the coefficient of discharge. 

Fteley and Stearns adopt the above formula for the dis- 
charge, or placing m for c . -| V2g, they write,** 

qi = mb{H+\H')(H-Hy; . . . (37)' 
and from their experiments deduce the following values of m : 

JJTI 

For -^ = 0.00 0.04 0.08 0.12 0.16 0.2 0.3 
^ = 3-33 3.35 3-37 3.35 3.32 3-28 3-2i 

For -jy — 0.4 0.5 
7// = 3.15 3. 11 

These are for suppressed weirs ; for contracted weirs few or no 

experiments are on record. 

In what has thus far been said velocity of approach has not 
been considered. This may be taken into account in the usual 
way by determining the velocity-head /i, and thus correcting 
H. In strictness the velocity of departure in the tail bay below 

* Transactions American Society Civil Engineers, 1SS3, vol. xii. p. 103. 



0.6 


0.7 


0.8 


0.9 


1.0 


3-09 


3-09 


3.12 


3.19 


3-33 



Art. 57.] 



ROUNDED AND WIDE CRESTS. 



117 



the weir should be regarded, and its head h! be applied to H' 
But it is unnecessary, on account of the limited use of sub- 
merged weirs, and the consequent lack of experimental data, to 
develop this branch of the subject. What has been given 
above will enable a probable estimate to be made of the dis- 
charge in cases where the water accidentally rises above the 
crest, and further than this the use of submerged weirs cannot 
be recommended. 

Prob. 74. Compute by two methods the discharge over a 
submerged weir when b = 8, H = 0.46, and H' = 0.22 feet. 




Article 57. Rounded and Wide Crests. 

When the inner edge of the crest of a weir is rounded, as at 
A in Fig. 34, the discharge is materially increased as in the case of 
orifices (Art. 44), or rather the coefficients of discharge become 
much larger than those given 



for the standard sharp crests. 
The degree of rounding influ- 
ences so much the amount of B 
increase that no definite values FlG - 34. 
can be stated, and the subject is here merely mentioned in order 
to emphasize the fact that a rounded inner edge is always a 
source of error. If the radius of the rounded edge is small, 
the sheet of escaping water leaves it at a point below the top 
(a in the figure), which has the practical effect of increasing the 
measured head by a constant quantity. The experiments of 
Fteley and STEARNS show that when the radius is less than 
one-half an inch, the discharge can be computed from the usual 
weir formula, seven-tenths of the radius being first added to 
the measured head H. 

Two wide-crested weirs with square inner corners are shown 
in Fig. 34, the one at B being of sufficient width so that the 



n8 



FLOW OF WATER OVER WEIRS. 



[Chap. V. 



descending sheet may just touch the outer edge, causing the 
flow to be more or less disturbed, while that at C has the sheet 
adhering to the crest for some distance. In both cases the 
crest contraction occurs, although water instead of air may fill 
the space above the inner corner. Fori? the discharge maybe 
equal to or greater than that of the standard weir having the 
same head H, depending upon whether the air has or has not 
free access beneath the sheet in the space above the crest. For 
C the discharge is always less than that of the standard weir 
with sharp crest. 

The following table is an abstract from the results obtained 
by Fteley and Stearns,* and gives the corrections in feet to 
be subtracted from the depths on a wide crest, like C in Fig. 
34, in order to obtain the depths on a standard sharp-crested 
weir which will discharge an equal volume of water. 

TABLE XIII. CORRECTIONS FOR WIDE CRESTS. 



Head 

on wide 

crest. 

Feet. 


Width of crest in inches. 


2 


4 


6 


8 10 


12 


24 


O.05 


0.010 


O.OO9 


0.009 


O.OO9 


0.009 


. 009 


.009 


.10 


.016 


.Ol8 


.017 


.017 


.017 


.017 


017 


.20 


.012 


.029 


.031 


.032 


• 033 


• 033 


034 


.30 




.030 


.041 


045 


.047 


.04S 


050 


.40 




.022 


.045 


•055 


.060 


.062 


066 


•50 




.006 


.041 


.060 


.069 


.074 


0S2 


.60 






.031 .059 .075 


.0S3 


097 


.70 






.017 .052 .075 


.0S9 


112 


.80 






. 000 . 040 


.071 


.091 


125 


. 9 








.027 


.062 


.0S9 


137 


I. OO 








.Oil 


.050 


.0S2 


1-49 


1.20 










.021 


.061 


168 


I.40 

1 












.032 


180 



* Transactions American Society Civil Engineers, 1SS3, vol. xii. 96. 



Art. 58]. WASTE WEIRS AND DAMS. I 1 9 

These results were obtained by passing a constant volume 
of water over a standard weir and measuring the head//" on the 
crest ; a piece of timber was then brought into place on the 
lower side of the crest and secured by fastenings, thus forming 
the wide crest ; and the head H being again measured, the in- 
crease of depth was thus obtained. This being repeated for 
different constant volumes the results were plotted and mean 
curves drawn, from which the table was derived. The weir 
used was without end contractions, and to such only the con- 
clusions apply with precision. For weirs with end contractions 
where the air has free access under the sheet at the ends the 
discharge is probably different. 

Prob. 75. Compute the discharge over a crest 1.5 feet wide 
for a weir 10 feet long when the head is 0.850 feet, and show 
that the discharge is about 19 per cent less than that over a 
standard sharp-crested weir under the same head. 

Article 58. Waste Weirs and Dams. 

Waste weirs are constructed at the sides of canals and 
reservoirs in order to allow surplus water to escape. They are 
usually made with wide crests, the inner approach to which is 
inclined, and the discharge is received upon an apron of timber 
or masonry. The flow over these wide-crested weirs is always 




Fig. 



much less than for equal depths on standard weirs, and for 
narrow crests the diminution may be approximately estimated 
by the use of the table in the preceding article. When 
the crest is about 3 feet wide, and level, with a rising slope 



120 FLOW OF WATER OVER WEIRS. [Chap. V. 

to its inner edge, and the end contractions are suppressed, 
the following formula, deduced by FRANCIS, may be applied, 

q = z.oibH**, 

in which b and H are to be taken in feet, and q is in cubic feet 

per second. 

In constructing a waste weir the discharge q is generally 
known or assumed, and it is required to determine b and H. 
The latter being taken at I, 2, or 3 feet, as may be judged safe 
and proper, b is found by 



3.0I77 1 - 53 



If, for example, q be 87 cubic feet per second, and H be taken 
as 2 feet, then 

log b - log 87 - log 3.01 - 1.53 log 2, 
from which 

log b = 1.0004, 

whence b = 10.0 feet. If, however, H be taken as 1 foot, b is 
required to be nearly 30 feet. 

The ordinary weir formula may be also used for waste-weir 
calculations with results differing but little from those obtained 
by the above expression. Or using the approximate general 
expression from Art. 55, 

In this, if q be 87 cubic feet per second, and H be 2 feet, the 
value of b is found to be 9.24 feet. Evidently no great pre- 
cision is needed in computing the length of a waste weir, since 



Art. 58.] 



WASTE WEIRS AND DAMS. 



121 



it is difficult to determine the exact discharge which is to pass 
over it, and ample allowance must be made for unusual rains 
or floods. 



When a dam is built across a stream it is often im 
to arrange its height so that 
the water level may stand 
at a certain elevation. In 
Fig. 36 the line^ CC repre- 
sents the surface of the 
stream before the construc- 
tion of the dam, the depth 
of water being D, and it is 
required to find the height 
of the dam G, so that the 
surface may be raised the 

distance d' . If the crest be FlG - 36 - 

not submerged, as in the first diagram, 

G = D 4- d' - II. 



portant 




In this H is to be inserted in terms of the discharge q, or the 
length b is to be determined as above for an assumed value of 
H. For the former method, 



G = £ + d / - 



* V 



3-33^ 



in which b may be width of the stream or less, as the design 
requires. If G, B, q, and b be given, this formula may be used 
to compute d' . 

If the height of the dam is small, as in the second diagram 
of Fig. 36, the crest is submerged, and the last formula will not 
apply. For this case 



H=D + d f - G, 



H' = D-G\ 



122 FLOW OF WATER OVER WEIRS. [Chap. V. 

and inserting these heads in the formula (37)', and solving for 
G, the following result is found : 

G = D + \d' - 2q 



ynb Vd' 

In this formula m lies between 3.09 and 3.37, depending on the 
value of the ratio H ' ' ■— H, and accordingly a tentative method 
of solution must be adopted. For example, let D = 4 feet, 
d' = 1 foot, b ■== 50 feet, and q = 400 cubic feet per second; 
then, assuming m as 3.33, 

G — 4 +0.67 — 1.6 = 3.1 feet. 

Now H= 4 + 1 — 3.1 = 1.9 feet, and H' = 4 — 3.1 .= 0.9, so 
that the ratio H' -=- H= 0.47, and hence from Art. 56 the 
value of m is about 3.12. Using this, the value of G is now 
computed to be 2.96 feet, which gives H = 2.04 feet, and 
H' = 1.04 feet, and H' -r- H — 0.5, which indicates that no 
further variation in 111 will be found. Accordingly 2.96 feet is 
the required height of the submerged dam. 

Prob. 76. If 150 cubic feet per second flow over a waste 
weir 20 feet long, find the depth of water on the crest. 

Prob. yy. A stream 4 feet deep which delivers 150 cubic feet 
per second is to be dammed so as to raise the water 6 feet 
higher. Find the height of the dam when the length of the 
overflow is 12 feet. 



Article 59. The Surface Curve. 

The surface of the water above a weir assumes during the 
flow a curve whose equation is not known, but some of the 
laws which govern it maybe deduced in the following manner: 
Let H be the head above the level of the crest measured in 



Art. 59.] 



THE SURFACE CURVE. 



™3 



perfectly level water at some distance back of the weir, and let 
d be the depression or drop of the curve 
below this level in the plane of the weir 
(Fig. 37). The discharge per second q 
can be expressed in terms of H and d by 
formula (n)' of Art. 25 by placing H for 
h 2 and d for h x . This, multiplied by a 
FlG - 37. coefficient k, gives, if velocity of approach 

be neglected, the formula 

q = k.\^2g.b{H\ — di). 




This expression, it may be remarked, is the true weir formula, 
and only the practical difficulties of measuring d prevent its 
use. 

From this formula the value of the drop d in the plane of 
the weir is found to be 



di = Hi - 



3? 



2kb V2g 



Let B be the breadth of the feeding canal, G its depth below 
the crest, and v the mean velocity of approach ; then 

q = B(G + H)v. 

3 v 
Inserting this in the equation, replacing —7- by m, and by 

its value h?, where h is the velocity-head corresponding to v y 
the formula becomes 

d% = H%-m^(G+~H)hK .... (38) 



which is an expression for the drop of the curve in terms of the 
dimensions of the feeding canal and weir, and the heads H 
and h. 



124 FLOW OF WATER OVER WEIRS. [Chap. V. 

The approximate value of the coefficient m is about 2.2, 
but precise values of d cannot be computed unless m and H 
are known with accuracy. The formula, however, serves to 
exemplify the laws which govern the drop of the curve in the 
plane of the weir. It shows that the drop increases with the 
head on the crest and with the length of a contracted weir, that 
it decreases with the breadth and depth of the feeding canal, 
and that it decreases with the velocity of approach. It also 
shows for suppressed weirs, where B = b, that the drop is inde- 
pendent of the length of the weir. All of these laws except 
the last have been previously deduced by the discussion of 
experiments. 

Prob. 78. Discuss the above formula when H = o; also 
when h = o. 



Article 60. Triangular and Trapezoidal Weirs. 

Triangular ribtches are used but little, as in general they are 
only convenient when the quantity of water to be measured is 
small. Such a notch when used as a weir must have sharp 
inner corners, so that the stream may be fully contracted, and 
the sides should have equal slopes. The angle at the lower 
vertex should be a right angle, as this is the only case for which 
coefficients are known with precision. The depth of water 
above this lower vertex is to be measured by a hook gauge in 
the usual manner at a point several feet up stream from the 
notch. 

In Art. 23 is deduced a formula for the theoretic discharge 
through a triangular notch. Making the angle at the vertex a 
right angle, and applying a coefficient, the actual discharge per 
second is given by the expression 



A t. 60.] TRIANGULAR AND TRAPEZOIDAL WEIRS. 1 25 

in which H is the head of water above the vertex. Experi- 
ments made by THOMSON* indicate that the coefficient c 
varies less with the head than for ordinary weirs; this, in fact, 
was anticipated, since the sections of the stream are similar in 
a triangular notch for all values of H, and hence the influence 
of the contractions in diminishing the discharge should be ap- 
proximately the same. As the result of his experiments the 
mean value of c for heads between 0.2 and 0.8 feet may be 
taken as 0.^92, and hence the mean discharge in cubic feet per 
second through a right-angled triangular weir may be written 

£=2.54^, 

in which, as usual, H must be expressed in feet. 

A trapezoidal weir has a similar advantage in rendering the 
coefficients nearly constant. The proportions recommended 
by CiPPOLETTl are that the slope of the ends should be 1 to 
4 whatever be the length of 
the sill /, or that, in the figure, 
z = \H. The reasoning from 
which this conclusion is derived 
is based upon FRANCIS' rule 
(Art. 55), that each end contrac- 
tion diminishes the discharge by a mean amount 3.33 X O.I X 
H*, or in general by the amount c X f V2g X 0.1 X H\ If 
the end be sloped, however, the discharge through the end tri- 
angle having the base z and depth H is (Art. 23) c X tV V2g X 
z X // ? - If now the slope is just sufficient so that the extra 
discharge balances the effect of the end contraction, these two 
quantities are equal. Equating them, and supposing that c has 
the same value in each, there results z == \H. Hence in such 
a trapezoidal weir the discharge should be the same as from 

* British Association Report, 1858, p. 133. 




126 FLOW OF WATER OVER WEIRS. [Chap. V. 

a suppressed rectangular weir of length /, or, according to 
Francis, q — 3.33/Zri Cippoletti, however, concluded from 
his experiments that the coefficient should be increased about 
one per cent, and wrote 

q= 3.367 I Hi 

as the formula for discharge when no velocity of approach 
exists. 

Recent experiments by Flinn and Dyer* indicate that the 
coefficient 3.367 is probably a little too large. In 32 tests with 
trapezoidal weirs of from 3 to 9 feet length on the crest and 
under heads ranging from 0.2 to 1.4 feet, they found 28 to give 
discharges less than the formula, the percentage of error 
being over 3 per cent in eight cases. The four cases in which 
the discharge was greater than that given by the formula show 
a mean excess of about 3.5 per cent. The mean deficiency in 
all the 32 cases was nearly 2 per cent. These experiments are 
not very precise, since the actual discharge was computed by 
measurements on a rectangular weir, so that the results are 
necessarily affected by the errors of two sets of measurements, 
as well as by leakage, which probably could not be wholly 
accounted for. ClPPOLETTi's formula, given above, may hence 
be allowed to stand as a fair one for general use with trapezoidal 
weirs.f It can, of course, be written in the form 

q = c.\VTglHK 

in which c has the mean value 0.629. 

If velocity of approach exists, // in this formula is to be re- 
placed by H -\- 1.4/1 where h is the head due to that velocity. 
In order to do good work, however, h should not exceed 0.004 

* Transactions American Society of Civil Engineers, July, 1S94, pp. 9-33. 
f Cippoletti, Canal Villoresi, 18S7; see Engineering Record, Aug. 13, 1S92. 



Art. 60.] TRIANGULAR AND TRAPEZOIDAL WEIRS. \2J 

feet. Other precautions to be observed are that the cross-sec- 
tion of the canal should be, at least, seven times that of the 
water in the plane of the crest, and that the error in the 
measured head should not be greater than one-third of one per 
cent. 

Prob. 79. For a head of 0.7862 feet on a ClPPOLETTI weir 
of 4 feet length the actual discharge in 420 seconds was 3912.3 
cubic feet. Compute the discharge by the formula and find 
the percentage of error. 



128 FLOW THROUGH TUBES. [Chap. YL 




CHAPTER VI. 
FLOW THROUGH TUBES. 

Article 6i. The Standard Short Tube. 

A standard tube is a very short pipe, whose length is 
about three times its diameter, or of sufficient length so that 
the escaping jet just fills its outer end, 
and there issues without contraction. 
The inner end of the tube is placed flush 
with the inner side of the reservoir, and 
is to be a sharp, definite corner, like that 
of the standard orifice (Art. 34). Fig. 38. 

The phenomena of flow through such a tube are similar in 
some respects to those of the flow from the standard orifice, 
but the discharge is much greater. By observations with glass 
tubes it is found that the contraction of the jet occurs as in the 
orifice, although agitation of the water or a shock upon the 
tube is apt to apparently destroy it, and cause the entire length 
to be filled. If, however, holes be bored in the tube near its 
inner end, water does not flow out, but air enters, showing that 
a negative pressure exists. 

Since the issuing jet entirely fills the outer end of the tube, 
the coefficient of contraction for that section is unity (Art. 35), 
and hence the coefficient of velocity equals the coefficient of 
discharge (Art. 37). Numerous experiments by VENTURI, 
BOSSUT, CASTEL, and others, give the following as a mean 
value for the standard tube : 

c = 0.82. 



Art. 6i.] 



THE STANDARD SHORT TUBE. 



129 



This value, however, ranges from 0.83 for low heads and small 
tubes to 0.80 for high heads and large tubes, its law of varia- 
tion being probably the same as for orifices (Art. 38), although 
experiments are wanting from which to state definite values in 
the form of a table. 

A standard orifice gives on the average about 61 per cent 
of the theoretic discharge, but by the addition of a tube this 
may be increased to 82 per cent. The effective energy of the 
jet from the tube is, however, much less than that from the 
orifice. For, let v be the velocity and h the head, then (Art. 
36) for the orifice 



v = 0.98 y2gk, whence — == 0.96/2 ; 

2g 



and similarly for the tube, 



0.82 V2gk f whence — = 0.67k. 
2 g 



Accordingly, the effective energy of the stream from the orifice 

is 96 per cent of the theoretic A: 

energy, while that of the 

stream from the tube is only 

67 per cent. Or if jets be 

directed vertically upward 

from a standard orifice and a 

standard tube, as in Fig. 39, 

that from the former rises to 

the height 0.96/2, while that 

from the latter rises to the 

height 0.67/2, where h is the 

head from the level of water AB in the reservoir to the point 

of exit. 




Fig. 



130 FLOW THROUGH TUBES. [Chap. VI. 

The standard tube is not used for the measurement of water, 
as this can be done with greater precision and convenience by 
orifices. It is important, however, to know the general laws of 
flow which have here been set forth, as a starting point in the 
theory of pipes, and for other purposes. The fact that the tube 
gives a greater discharge than an orifice is an interesting one, 
and the reason for this will be explained in Art. 67. 

Prob. 80. Compare the effective horse-power of the streams 
from a standard orifice and tube, the diameter of each being 
4 inches and the head 25 feet. 

Article 62. Conical Converging Tubes. 

Conical converging tubes are used when it is desired to 
obtain a high efficiency in the energy of the stream of water. 

At A is shown a simple con- 
verging tube, consisting of a 
= frustum of a cone, and at B 
is a similar frustum, provided 
with a cylindrical tip. The 
Fic - 40. proportions of these converg- 

ing tubes, or mouthpieces, vary somewhat in practice, but the 
cylindrical tip when employed is of a length equal to about 
2^ times its inner diameter, while the conical part is eight or 
ten times the length of that diameter, the angle at the vertex 
of the cone being between 10 and 20 degrees. 

The stream from a conical converging tube like A suffers a 
contraction at some distance beyond the end. The coefficient 
of discharge is higher than that of the standard tube, being 
generally between 0.85 and 0.95, while the coefficient of velocity 
is higher still. Experiments made by D'AUBUISSON and CASTEL 
on conical converging tubes 0.04 meters long and 0.0155 meters 
in diameter at the small end, under a head of 3 meters, give 




Art. 62.] 



CONICAL CONVERGING TUBES. 



131 



the following results for the coefficients of discharge and 
velocity, the former being determined by measuring the actual 
discharge (Art. 37), and the latter by the range of the jet (Art. 
36). The coefficient of contraction, as computed from these, 
is given in the last column ; and this applies to the jet at the 
smallest section, some distance beyond the end of the tube. 

TABLE XIV. COEFFICIENTS FOR CONICAL TUBES. 



Angle of Cone. 


Discharge 
c. 


Velocity 
c x . 


Contraction 
c' . 


o° 00' 


0.829 


O.829 


1.00 


I 36 


O.866 


O.867 




4 IO 


O.912 


O.9IO 




7 52 


O.93O 


O.932 


O.998 


10 20 


O.938 


O.951 


O.986 


13 24 


O.946 


O.963 


O.9S3 


16 36 


O.938 


O.971 


O.966 


21 00 


O.919 


O.972 


0-945 


29 58 


O.895 


0-975 


O.918 


48 50 


O.847 


984 


O.861 



While these values show that the greatest discharge occurred 
for an angle of about 13^- degrees, they also indicate that the 
coefficient of velocity increases with the convergence of the 
cone, becoming about equal to that of a standard orifice for 
the last value. Hence the table seems to teach that a conical 
frustum is not the best< form for a mouthpiece to give the 
greatest velocity. 

Under very high heads — over 300 feet — SMITH found the 
actual discharge to agree closely with the theoretical, or the 
coefficient of discharge was nearly 1.0, and in some cases slightly 
greater.* His tubes were about 0.9 feet long, 0.1 feet in 



* Smith's Hydraulics, p. 286. 



132 FLOW THROUGH TUBES. [Chap. VL 

diameter at the small end and 0.35 feet at the large end, the 
angle of convergence being 17 degrees. As this implies a con- 
traction of the jet beyond the end, it cannot be supposed that 
the coefficient of discharge in any case was really as high as 
his experiments indicate. Under these high heads the cylin- 
drical tip applied to the end of a tube produced no effect on 
the discharge, the jet passing through without touching its 
surface. 

Prob. 81. If the coefficient of discharge is 0.98 and the 
coefficient of velocity 0.995, compute the coefficient of con- 
traction. 

Article 63. Nozzles and Jets. 

For fire service two forms of nozzles are in use. The smooth 
nozzle is essentially a conical tube like A in Fig. 40, the larger 
end being attached to a hose, but it is 
!§p often provided with a cylindrical tip 
and sometimes the inner end is curved 
as seen in the upper diagram of Fig. 41. 
The ring nozzle is a conical tube having 
an orifice whose diameter is slightly 
smaller than that of the end of the tube. 
FlG " 41 ' The experiments of FREEMAN show 

that the mean coefficient of discharge is about 0.97 for the 
smooth nozzle and about 0.74 for the ring nozzle* They also 
seem to indicate that the simple cone has a higher discharge 
than any form of curved nozzle. 

The effective head at the entrance to a nozzle is the pres- 
sure-head plus the velocity-head (Art. 27). Let D be the 
diameter of the pipe or hose, dTthe diameter of the outlet end 
of the nozzle, and Fand v the corresponding velocities. Let 





* Freeman, The Hydraulics of Fire Streams. Transactions American Society 
of Civil Engineers, 1SS9, vol. xxi. pp. 303-4S2. 



Art. 63.] NOZZLES. 133 

h x be the pressure-head at the entrance ; then the effective head 
at the entrance of the nozzle is 

V 2 

2g 

and the velocity of discharge is v = c x V2gH. Also VD 1 = vd*, 
since the same quantity of water per second passes the two 
sections. From these two equations the values of //and Fcan 
be expressed- in terms of v 9 and inserting them in the formula 
and solving for v there is found 



for the velocity of discharge from 'the nozzle. Here the last 
term in the denominator shows the effect of the velocity of ap- 
proach in the pipe ; if c 1 = 1, it agrees with the theoretic ex- 
pression deduced in Art. 25. In order to use this formula h x 
must be measured by a pressure-gauge at the entrance to the 
nozzle ; if this gives the pressure p 1 in pounds per square inch, 
then h l = 2.304^ (Art. 9). For smooth nozzles where there 
is no contraction of the stream after exit the coefficient of 
velocity c l is equal to the coefficient of discharge c. 

The effective head at the entrance to the nozzle may now 
be written 

H "■ 



and the effective head at the orifice of the nozzle is 

1 - c \d) 



134 FLO IV THROUGH TUBES. [Chap. VI. 

which gives the height to which the jet would rise if there 
were no atmospheric resistances. The discharge is the product 
of the area of the orifice and the velocity, or \nd k v, and hence 



q = 6.2 99 c 1 d\/ '-i— - .... (40) 

gives the discharge in cubic feet per second. 

The experiments of FREEMAN furnish the following mean 
values of the coefficient of discharge for smooth cone nozzles 
of different diameters under pressure-heads ranging from 45 to 
180 feet: 

Diameter = f, -J, I, ij, ij, if inches 

Coefficient c = 0.983, 0.982, 0.972, 0.976, 0.971, 0.959 

These values were determined by measuring the pressure- 
head h l , and the discharge q, from which c y can then be com- 
puted by (40), its value in this case being the same as c. 
For example, a nozzle whose diameter was 1.001 inches at the 
orifice and 2.5 inches at the base discharged 208.5 gallons per 
minute under a pressure of 50 pounds at the entrance ; here d = 
1. 001/12, D — 2.5/12, h x = 50 X 2.304, q = 208.5 X O.1337/60, 
and inserting all quantities in (40) and solving for c x there is 
found c 1 = O.985. 

The vertical height of a jet from a nozzle is very much less, 
on account of the resistance of the air, than the value given by 
(39)'. For instance, let a smooth nozzle one inch in diameter 
attached to a 2.5-inch hose have ^ = 0.97 and the pressure- 
head h x = 230 feet ; then (39)' gives h = 221 feet, whereas the 
average of the highest drops in still air will be about 152 feet 
high and the main body of water will be several feet lower. 
The following table, compiled from the results of Freeman's 
experiments, shows for three different smooth nozzles, the 



Art. 63.] NOZZLES. 1 35 

TABLE XV. VERTICAL JETS FROM SMOOTH NOZZLES. 



Indicated 


From %-inch Nozzle. 


From 


i-inch Nozzle. 


From 


i|-inch Nozzle. 


Pressure at 
Entrance to 

Nozzle. 
Pounds per 

Square 
Inch. 




















Height 
A 


in Feet. 
B 


Dis- 
charge. 
Gallons 

per 
Minute. 


Height 


in Feet. 


Dis- 
charge. 
Gallons 

per 
Minute. 


Height 


in Feet. 


Dis- 
charge. 
Gallons 

per 
Minute. 


A 


B 


A 


B 


10 


20 


17 


52 


21 


iS 


93 


22 


19 


148 


20 


40 


33 


73 


43 


35 


132 


44 


37 


209 


30 


59 


48 


90 


63 


5i 


161 


66 


53 


256 


40 


73 


60 


104 


83 


64 


1S6 


86 


67 


296 


50 


93 


67 


116 


101 


73 


208 


107 


77 


331 


60 


104 


72 


127 


117 


79 


228 


126 


85 


363 


70 


114 


76 


137 


130 


85 


246 


140 


91 


392 


80 


123 


79 


147 


140 


89 


263 


150 


95 


419 


90 


129 


81 


156 


147 


92 


279 


157 


99 


444 


IOO 


134 


83 


164 


152 


96 


295 


161 


101 


46S 



height of vertical jets, column A giving the heights reached by 
the average of the highest drops in still air, and column B the 
maximum limits of height as a good effective fire-stream with 
moderate wind. The discharges given depend only on the 
pressure, and are the same for horizontal as for vertical jets. 

In ring nozzles the ring which contracts the entrance is 
usually only Jg or -§- inch in width. The effect of this is to 
diminish the discharge, but the stream is sometimes thrown to 
a slightly greater height. On the whole, ring nozzles seem to 
have no advantage over smooth ones for fire purposes. As 
the stream contracts after leaving the nozzle, the coefficient of 
velocity c x is greater than the coefficient of discharge c. The 
value of c being about 0.74, that of c x is probably a little larger 
than 0.97 in the contracted section. 

According to Freeman's experiments, the discharge of a 
■J-inch ring nozzle is the same as that of a f-inch smooth 
nozzle, while the discharge of a 1 4-inch ring nozzle is about 20 



I36 FLOW THROUGH TUBES. [Chap. VI. 

per cent greater than that of a i-inch smooth nozzle. The 
heights of vertical jets from a ij-inch ring nozzle are about 
the same as those from a i-inch smooth nozzle, while the jets 
from a if-inch ring nozzle are slightly less in height than those 
from a ij-inch smooth nozzle. 

The maximum horizontal distance to which a jet can be 
thrown is also a measure of the efficiency of a nozzle. The 
following, taken from Freeman's tables, gives the horizontal 
distances at the level of the nozzle reached by the average of 
the extreme drops in still air : 

Pressure at nozzle entrance, 
From f-inch smooth nozzle, 
From i-inch smooth nozzle, 
From i^-inch smooth nozzle, 
From j-^-inch ring nozzle, 
From r^-inch ring nozzle, 
From if-inch ring nozzle, 

The practical horizontal distances for an effective fire-stream is, 
however, only about one-half of these figures. 

The question as to the best form of curve for a nozzle, in 
order that the velocity may be a maximum, has often been 
discussed. In reality, however, no one curve has any advan- 
tage over others, for a high efficiency of the jet is secured only 
through avoiding the losses of energy. 

Prob. 82. A nozzle if inches in diameter attached to a play- 
pipe 2\ inches in diameter discharges 310.6 gallons per minute 
under an indicated pressure of 30 pounds per square inch. 
Find the effective head at the end of the nozzle and the co- 
efficient of velocity. 

Prob. 83. Find from the table the heights of vertical jets 
for a finch and a i-|-inch nozzle, and the discharges in gallons 
per minute, when the indicated pressure at the entrance is 75 
pounds per square inch. 



20 


40 


60 


80 


100 pour 


72 


112 


136 


153 


167 feet. 


77 


133 


167 


189 


205 feet. 


83 


148 


1S6 


213 


236 feet. 


76 


131 


164 


186 


202 feet. 


73 


138 


172 


196 


215 feet. 


79 


144 


1S0 


206 


227 feet. 



Art. 64.] 



DIVERGING AND COMPOUND TUBES. 



137 




Article 64. Diverging and Compound Tubes. 

In Fig. 42 is shown a diverging conical tube BC, and two 
compound tubes. The compound tube ABC consists of two 
cones, the converging one, AB, being much shorter than the 
diverging one, BC, so that the 
shape roughly approximates to 
the form of the contracted jet 
which issues from an orifice in 
a thin plate. In the tube AE 
the curved converging part AB 
closely imitates the contracted 
jet, and BB is a short cylinder 
in which all the filaments of 
the stream are supposed to 
move in lines parallel to the 
axis of the tube, the remaining 
part being a frustum of a cone. • The converging part of a 
compound tube is often called a mouthpiece, and the diverging 
part an adjutage. 

Many experiments with these tubes have shown the interest- 
ing and phenomenal fact that the discharge and the velocity 
through the smallest section, B, are greater than those due to the 
head ; or, in other words, that the coefficients of discharge and 
velocity are greater than unity. One of the first to notice this 
was BERNOULLI in 1738, who found c = 1.08 for a diverging 
tube. VENTURI in 1791 experimented on such tubes, and 
showed that the angle of the diverging part, as also its length, 
greatly influenced the discharge. He concluded that c would 
have a maximum value of 1.46 when the length of the diverg- 
ing part was 9 times its least diameter, the angle at the vertex 
of the cone being 5 06' '. Eytelwein found c — 1. 18 for a 
diverging tube like BC in Fig. 42, but when it was used as an 



138 FLOW THROUGH TUBES. [Chap. VI. 

adjutage to a mouthpiece, AB, thus forming a compound tube 

ABC, he found c = 1.55. 

The experiments of FRANCIS in 1854 on a compound tube 
like ABCDE are very interesting.* The curve of the converg- 
ing part AB was a cycloid, BB was a cylinder, and the diameters 
at A, B, etc., were 

A = 1.4 feet, (7 = 0.1454, E = 0.3209 

B — 0.1018, D = 0.2339, 

The piece BB was 0.1 feet long, and the others each 1 foot; 
these were made to screw together, so that experiments could 
be made on different lengths. A sixth piece, EF, not shown 
in the figure, was also used, which was a prolongation of the 
diverging cone, its largest diameter being 0.4085 feet. The 
tubes were of cast-iron, and quite smooth. The flow was 
measured with the tubes submerged, and the effective head 
varied from about 0.01 to 1.5 feet. Excluding heads less than 
0.1 feet, the following shows the range in value of the coeffi- 
cients of discharge : 





c for Section BB. 


c for Outer End. 


For tube AB, 


0.80 to 0.94 


0.80 to 0.94 


For tube AC, 


I.43 to 1.59 


0.70 to 0.78 


For tube AD, 


1.98 to 2.16 


0.37 to 0.41 


For tube AE, 


2.08 to 2.43 


0.21 tO O.24 


For tube AF, 


2.05 to 2.42 


0.13 to 0.15 



The maximum discharge was thus found to occur with the 
tube AE, and to be 2.43 times the theoretic discharge. In 
general the coefficients increased with the heads, the value 2.08 
being for a head of 0.13 feet and 2.43 for a head of 1.36 feet -, 
under 1.39 feet, however, c was found to be 2.26. 

The value of g at Lowell, Mass., where these experiments 

* Lowell Hydraulic Experiments, 4th Edition, pp. 209-232. 



Art. 64.] DIVERGING AND COMPOUND TUBES. 139 

were made, is about 32.162 feet per second. Hence under a 
head of 1.36 feet the theoretic velocity is 

V2gh = 8.0202 V1.36 = 9.36 feet per second, 
while the actual velocity in the section BB was 

v = 2.43 X 9.36 = 22.74 feet per second. 
The velocity-head corresponding to this is 

— = (2.43)7/ = 5..90*. 

Therefore the flow through the section BB was that due to a 
head 5.9 times greater than the actual head of 1.36 feet ; or, in 
other words, the energy of the water flowing in BB was 5.9 
times the theoretic energy. Here, apparently, is a striking 
contradiction of the fundamental law of the conservation of 
energy. 

Under high heads the velocity becomes so great that the 
jet does not touch the sides of the diverging tube, or adjutage, 
and hence the actual may not exceed the theoretic discharge. 
It is probable, however, that if the tube be long and its taper 
very slight an increased discharge can be obtained under 
a high head. 

The explanation of the phenomena of increased velocity 
and discharge caused by these tubes is simple. It is due to 
the occurrence of a partial vacuum near the inner end of the 
adjutage BC. The pressure of the atmosphere on the water 
in the reservoir thus increases the hydrostatic pressure due to 
the head, and the increased flow results. The energy at the 
smallest section is accordingly higher than the theoretic 
energy, but the excess of this above that due to the head must 
be expended in overcoming the atmospheric pressure on the 
outer end of the tube, so that in no case does the available ex- 



I40 FLOW THROUGH TUBES. [Chap. VI. 

ceed the theoretic energy. No contradiction of the law of 
conservation therefore exists. 

To render this explanation more definite, let the extreme 
case be considered where a complete vacuum exists near the 
inner end of the adjutage, if that were possible, as it perhaps 
might be with a tube of a certain form. Let h be the head of 
water in feet on the centre of the smallest section. The mean 
atmospheric pressure on the water in the reservoir is equivalent 
to a head of 34 feet (Art. 4). Hence the total head which 
causes the discharge into the vacuum is h -\- 34 and the 
velocity of flow is nearly V2g(/i -\- 34). Neglecting the re- 
sistances, which are very slight if the entrance be curved, the 
coefficients of velocity arid discharge can now be found ; thus : 



For h — 100, v — \2g X 134 = i- 16 V 2gh \ 
For h = 10, v = \ f 2g X 44 = 2.10 V2gh ; 



For h = 1, v — V 2 g X 35 = 5-9 2 v*gk- 
The coefficient hence increases as the head decreases. That 
this is not the case in the above experiments is undoubtedly 
due to the fact that the vacuum was only partial, and that the 
degree of rarefaction varied with the velocity. The cause of the 
vacuum, in fact, is to be attributed to the velocity of the 
stream, which by friction removes a part of the air from the 
inner end of the adjutage. 

It follows from this explanation that the phenomena of in- 
creased discharge from a compound tube could not be pro- 
duced in the absence of air. The experiment has been tried 
on a small scale under the receiver of an air-pump, and it was 
found that the actual flow through the narrow section dimin- 
ished the more complete the rarefaction. It also follows that it is 
useless to state any value as representing, even approximately, 
the coefficient of discharge for such tubes. To secure the high- 
est coefficients, it is thought that the form of the adjutage of 



Art. 65.] 



INWARD PROJECTING TUBES. 



141 



Fig. 



43- 



the compound tube should not be conical, but of the shape de- 
duced for the perfect nozzle in Art. 63. The converging part 
should also properly be of the 

same form. Then the stream ~~\ __^ /— 

both in contracting and in ex- 
panding follows the law of 
the perfect jet ; and hence it 
may be supposed that the least loss of energy will result, and 
consequently the greatest flow. This, however, is a mere 
hypothesis, not yet confirmed by experiment. 

Prob. 84. Compute the pressure per square inch in the 
section BB of Francis' tube when h = 1.36 feet and c = 2.43. 
What is the height of the column CD (Fig. 19, Art. 27) that 
could be lifted by a small pipe inserted at BB} 

Article 65. Inward Projecting Tubes. 

Inward projecting tubes, as a rule, give a less discharge 
than those whose ends are flush with the sides of the reser- 
voir, due to the greater convergence of the lines of direction 
of the filaments of water. At A and B are shown inward pro- 
jecting tubes so short that the water merely touches their inner 
edges, and hence they may more properly be called orifices. 
Experiment shows that the case at A, where the sides of the 
tube are normal to the side 



C=0.50 



'■MfcA. b=-mv=^ 



H A HI BJlfc 



:c 5:0:72: 



of the reservoir, gives the 
minimum coefficient of dis- 
charge c = 0.5, while for B 
the value lies between 0.5 
and that for the standard 
orifice at C. The inward 
projecting cylindrical tube 
at D has been found to give 
a discharge of about 72 per cent of the theoretic discharge, 
while the standard tube (Art. 61) gives 82 per cent. For the 




Fig. 



142 FLOW THROUGH TUBES. [Chap. VI. 

tubes E and F the coefficients depend upon the amount of 
inward projection, and they are much larger than 0.72 for 
both cases, when computed for the area of the smaller end. 

It is usually more convenient to allow a water-main to pro- 
ject inward into the reservoir than to arrange it with its mouth 
flush to a vertical side. The case D, in Fig. 44, is therefore of 
practical importance in considering the entrance of water into 
the main. As the end of such a main has a flange, forming a 
partial bell-shaped mouth, the value of c is probably higher 
than 0.72. The usual value taken is 0.82, or the same as for 
the standard tube (Art. 61). Practically, as will be seen in a 
later article, it makes little difference which of these is used, 
as the velocity in such a pipe is slow and the resistance at the 
mouth is very small compared with the frictional resistances 
along its length. 

Prob. 85. Find the coefficient of discharge for a tube whose 
diameter is one inch, when the flow under a head of 9 feet is 
22.1 cubic feet in 3 minutes and 30 seconds. 

Article 66. Effective Head and Lost Head. 

The terms energy and head are often used as equivalent, 
although really energy is proportional to head. Thus, if h be 

the head on an orifice or tube, — the velocitv head of the issu- 

Ocr 

ing jet, and IF the weight of water discharged per second, the 

theoretic energy per second is W/i, the effective or actual en- 

T' 2 / 7-' \ 

ergy is W — , and the lost energy is W\h J . It is more 

2g \ 2g 7 

convenient to deal directly with the heads, omitting the W\ 
thus the effective head in this case is — , and the lost head is 

,_,2 

h — — \ 
*g 



Art. 66.] EFFECTIVE HEAD AND LOST HEAD. 143 

If no losses occur due to friction, contraction, or other 
causes, the effective head at any point of a tube or pipe is 
equal to the hydrostatic head h. This effective head may be 
exerted either in producing pressure or in producing velocity, 
or part of it in pressure and part in velocity. Thus, as shown 

in Art. 27, 

v 2 

where h x is the pressure-head at the place considered. If 
there be no motion of the water h equals //, , and if the flow is 

...2 

so rapid that there be no pressure h equals — . Owing to the 

-.2 

various resistances, however, the effective head h. -\- - — is gen- 

erally less than the total head h, and the difference is called the 
lost head. Thus, at any section of a tube or pipe the head 
which has been lost is 



' =/i -^+B (4.) 



At the end of the tube, or rather outside of the tube, there 
can be no pressure on the jet, and the loss of head in the flow 
of the jet hence is 



v~ 



h' = h--. ...... (41)' 

Thus in Art. 46 it was shown that for the standard orifice 
the loss of energy or head is about 4 per cent, and in Art. 61 it 
was shown that for the standard tube the loss is about 33 
per cent. 

In any case the loss of head in a jet from a tube or orifice 
depends merely on the loss of velocity. Let c x be the coeffi- 
cient of velocity : then for a small orifice or tube 



144 FLOW THROUGH TUBES. [Chap. VI. 

and the effective velocity-head is 

Consequently the loss of head is 

h' = k-^={i-cS)k (42) 

It is sometimes more convenient especially for pipes to express 
this loss in terms of the velocity-head. The value of h in terms 
of this is 

and hence the loss of head is 

*-(?-)£ «■* 

in which v is the actual velocity of discharge. 

For the standard tube (Fig. 38, Art. 61) the coefficient of 
velocity is equal to the coefficient of discharge whose mean 
value is 0.82. The effective head of the jet then is 

— = (0.82) 9 /* = 0.67/*, 
2 g 

and the loss of head is 

ti = (1 _ 0.67)// = 0.33/;, 
or 

„ / 1 \ v 9 

•49 



VO.67 J 2g 



*g 

Hence the loss of head may be said to be either 33 per cent 
of the total head or 49 per cent of the effective velocity-head ; 
that is, the lost energy is about one-third of the total energy 
or about one-half of the effective energy. 



Art. 67.] LOSSES IN THE STANDARD TUBE. 145 

In reality, work or energy is never lost, but is merely trans- 
formed into other forms of energy. In the tube the one-third 
of the total energy which has been called lost is only lost 
because it cannot be utilized as work ; it is, in fact, transformed 
into heat, which raises the temperature of the water. And so 
it is in all cases of lost head : the pressure-head plus the 
velocity head is the effective head which can alone be rendered 
useful; if, this be less than the total hydrostatic head, the 
remainder has disappeared in heat. 

Prob. 86. Show that the lost head is nearly equal to the 
effective head for an inward projecting cylindrical tube. 

Article 67. Losses in the Standard Tube. 

The loss of head in the flow from the short cylindrical tube 
is large, but not so large as might be expected from theoretical 
considerations based on the known coefficients for orifices. 
If the tube has a length of only two diameters the jet does 
not touch its inner surface, and the flow occurs as from a 
standard orifice. The velocity in the plane of the inner end 
is then 61 per cent of the theoretic velocity, since the mean 
coefficient of discharge is 0.61. Now if the tube be increased 
in length about one diameter its outer end is filled by the jet, 
and since the contraction still exists, it might be inferred that 
the coefficient for that end would be also 0.61 : this would 
give an effective head of (o.6i)Vz or 0.37//, so that the loss of 
head would be 0.63/^. Actually, however, the coefficient is 
found to be 0.82 and the loss of head only 0.33//. It hence 
appears that further explanation is needed to account for the 
increased discharge and energy. 

It is to be presumed, in the first place, that a loss of about 
0.04/^ occurs at the inner end of the tube in the same manner 
as in the standard orifice, due to retardation of the outer fila- 
ments (Art. 46). The effective head at the contracted section 



146 



FLOW THROUGH TUBES. 



[Chap. VI. 



in the tube is then about 0.96//. If the coefficient of contrac- 
tion have the value 0.62, as in the orifice, the velocity in that 
section is greater than at the end of the tube, and, since the 
velocities are inversely as the areas of the sections, that velo- 
city is 

0.82, — 7 / — - 

v i = ^ V 2 £ h = !-32 V2gh, 

which is nearly one-third larger than the theoretic velocity. 
The velocity-head at that section then is 



and consequently the pressure-head is 

k x = 0.96/1 — i./$/i — — 0.79J1. 

There exists therefore a negative pressure or partial vacuum 
in the tube which is sufficient to lift a column of water to a 
height of about three-fourths the head. 
This conclusion has been confirmed by 
experiment for low heads, and was in 
fact first discovered experimentally by 
H VENTURI. For high heads it is not 
valid, since in no event can atmospheric 
pressure raise a column of water higher 
than about 34 feet (Art. 4) ; probably 
under high heads the coefficient of con- 
- traction of the jet in the tube becomes 



fi- 



ll 1 



Fig. 45. 



much greater than 0.62. 



The reason of the increased discharge of the tube over the 
orifice is hence due to the negative pressure or partial vacuum, 
which causes a portion of the atmospheric head of 34 feet to 
be added to the head //, so that the flow at the contracted 
section occurs as if under the head h -j- /^ , as in the diverging 



Art. 67.] LOSSES IN THE STANDARD TUBE. 147 

tube (Art. 64). The occurrence of the partial vacuum is attrib- 
uted to the friction of the sides of the jet on the air. When 
the flow begins, the jet is' surrounded by air of the normal 
atmospheric pressure which is imprisoned as the jet fills the 
tube. The friction of the moving water carries some of this 
air out with it, thus rarefying the remaining air. This rarefac- 
tion, or negative pressure, is followed by an increased velocity 
of flow, and the process continues until the air around the con- 
tracted section is so rarefied that no more is removed, and the 
flow then remains permanent, giving the results ascertained by 
experiment. The experiments of BUFF have proved that in 
an almost complete vacuum the discharge of the tube is but 
little greater than that of the orifice.* 

The velocity-head in the contracted section of the jet is thus 
about 1.75/2, but of this 0.79/2 must be expended in overcom- 
ing the atmospheric pressure at the end of the tube, so that 
the effective head is only 0.96//. If the retarding influence of 
the outer end be 0.04/2, or the same as that of the inner end, 
the effective head is reduced to 0.92/2, while the actual effect- 
ive velocity-head is 0.67/2. Thus a further loss of 0.25/2 is to 
be accounted for, and this must be supposed to be due to the 
enlargement of the section of the jet, and the consequent dimi- 
nution of velocity, whereby the energy is converted into heat. 
The partial vacuum causes neither a gain nor loss of head, and 
the only losses are 0.04/2 at the inner end of the tube, 0.25/2 
in the enlargement of the jet, and 0.04/2 at the outer end, or 
in all 0.33/2. These quantities, of course, are only approxi- 
mate, as they depend upon the mean coefficients 0.98, 0.62, 
and 0.82, all of which are liable to variation. 

Prob. 87. Discuss the losses of head in an inward projecting 
tube, taking c = 0.6 and c = 0.7. 

* See Ruhlmann's Hydromechanik (Hannover, 1879). 



148 



FLOW THROUGH TUBES. 



[Chap VL 



Article 68. Loss due to Enlargement of Section. 

When a tube or pipe is kept constantly full of water a loss 
of head is found to result when the section is enlarged so that 

the velocity is diminished. Let 
v 1 and z' 2 be the velocities in the 
smaller and larger sections, and 
h x and /* 2 the corresponding pres- 
sure-heads. The effective head 
in the first section is the sum of 
the pressure- and velocity-heads 
(Arts. 27 and 66), or 




Fig. 46. 






and the effective head in the second section is 

2g 

If no losses occur, these two expressions are equal ; but as the 
second effective head is always smaller than the first, their dif- 
ference is the loss of head between the two sections, or the 
lost head h' is 



ti 



^ - (//. - K). 



(43) 



This is a general expression, which gives the loss of head due 
not only to enlargement, but to all resistances between any two 
sections of a horizontal tube or pipe. If the difference //„ — Ji x 
of the pressure columns shown in Fig. 46 is measured, and the 
velocities determined, the loss of head is thus found in any 
particular case. 



The loss of head due to the sudden enlargement of section. 



Art. 68.] LOSS DUE TO ENLARGEMENT OF SECTION. 149 

or rather to the sudden diminution of velocity caused by the 
enlargement, can be expressed by the for- 
mula ■ 



*g 



To prove this, let p x be the unit pressure 
in AB and p^ that in CD. At a section 
MN very near the place of enlargement 




N D 

Fig. 47. 

the unit pressure is also^ , since the velocity v x is maintained 

for a short distance after leaving AB, its direction, however, 

being changed so as to form eddies. Let a 2 be the area of 

the section CD or MN. Then the pressure which acts in the 

opposite direction to the flow is ajp 2 — p x \ and this is the 

force which causes the velocity to diminish from v 1 to %\. Now 

in Art. 32 it was shown that the force which causes Impounds 

Wv 
of water to increase in velocity from o to v is , and con- 

versely the same force applied in the opposite direction will 
cause the velocity to diminish from v to o. Therefore the 
value of the pressure a 2 (p 9 — p x ) is 

*.(A - A) = — fa - O = — ~-z > 

o <5 

where w is the weight of a cubic unit of water. This expres- 
sion may be written, 

A _ vfa - v>) 



£1 

w 



w 



g 



or (Art. 9) 

This value of h^ 
reduces it to 



K _ K = ^_^3) . 
g 

k x inserted in the general equation (43) 
fa ~ *,Y 



hr = 



*g 



(44) 



I50 FLOW THROUGH TUBES. [Chap. VI. 

which is the formula for loss of head due to sudden enlarge- 
ment. The loss of energy in this case is similar to that which 
occurs in the impact of inelastic bodies, work being converted 
into heat. 

Let a x and <2 2 be the areas of the cross-sections AB and CD* 
Then v. = — -v. , and the formula for loss of head in sudden en- 
largement becomes 

*-e-.o> <«>■ 

which is often a more convenient form for practical use. If 
a x = a 2 or if v^ = o no loss of head results. 

If a gradual enlargement of section be made so that no im- 
pact occurs, the energy due to the velocity v x is slowly changed 
into pressure, so that head is not lost. There is, however, no 
distinct line of division between sudden and gradual enlarge- 
ment, and for a case like Fig. 46 experiment can alone deter- 
mine the value of h x — h^ and the loss of head. In the last 
article it was seen that about o.2$/i is lost in the expansion of 
the jet between the contracted section and the end of the tube. 
This seems like a case of gradual enlargement, but as no pres- 
sure can exist at the end of the tube the loss of head must 
be the same as for sudden enlargement of section ; in fact 
v x = 1.32 V2gh and v„ = 0.82 V2gh, whence by the above 
formula h! — 0.2 5 h. 

The loss of head due to sudden enlargement may often be 
very great, as the following example will show. Let the effec- 
tive head in the section AB be /i, all of which exists as velocity, 
so that v x = \ / 2gJi ; let the diameter of AB be 2 inches, and 
that of CD be 4 inches, so that the area at CD is four times 



Art. 69.] LOSS DUE TO CONTRACTION OF SECTION. 



151 



that at AB, and hence the velocity in CD is v 2 = \ VigL 
loss of head then is 



The 



/*' = 



ig 



9 // 



so that more than half the energy of the water in AB is lost in 
shock or impact. At CD the effective head is then -^k, of 
which -^h is velocity-head and -^h is pressure-head. Sudden 
enlargement of section is therefore to be avoided. 

Prob. 88. In a horizontal tube like Fig. 46 the diameters are 
6 inches and 12 inches, and the heights of the pressure-columns 
or piezometers are 12.16 feet and 12.96 feet above the same 
bench mark. Find the loss of head between the two sections 
when the discharge is 1.57 cubic feet per second, and also when 
it is 4.71 cubic feet per second. 



Article 69. Loss due to Contraction of Section. 

When a sudden contraction of section in the direction of 
the flow occurs, as in Fig. 48, the water suffers a contraction 
similar to that in the standard tube, and hence in its expansion 
to fill the smaller section a loss of head 
results. Let v x be the velocity in the 
larger section and v that in the smaller, 
while 1/ is the velocity in the contracted 
section of the flowing stream ; and let a x , 
a, and a' be the corresponding areas of 
the cross-sections. From the formula 
(44)' of the last article the loss of head 
due to the expansion of section from a' 
to a is 




Fig. 



W ) 1g 



2g 



(45) 



in which c' is the coefficient of contraction or the ratio of a! 
to a. 



152 



FLO W THROUGH TUBES. 



[Chap. VI. 



The value of c r depends upon the ratio between the areas 
a and a x . When a is small compared with a lt the value of c 
may be taken at 0.62 as for orifices (Art. 35). When a is 
equal to a 1 there is no contraction or expansion of the stream, 
and c is unity. Let d and d x be the diameters corresponding 
to the areas a and a lt and let r be the ratio of d to d 1 . Then 
experiments seem to indicate that an expression of the form 



c = m 



.1 — r 



gives the law of variation of c with r. Determining the values 
of m and n from the two limiting conditions above stated, 
there is found, 

0.0418 



c' = 0.582 + 



1.1 



from which approximate values of c can be computed. The 
manner of the variation in the values of c is indicated by the 
following" tabulation : 



For r = 0.0, 0.2, 0.4, 0.6, 0.7, 0.8, 0.9, 1.0, 

c' = 0.62, 0.63, 0.64, 0.67, 0.69, 0.72, 0.79, 1. 00. 

from which intermediate values may often be taken without 
the necessity of using the formula. 



For 
in Fig. 



a case 

49, the 




of gradual contraction of section, such as shown 
loss of head is less than that given by the above 
formula, and can only be de- 
termined for a given velocity of 
flow by observing the difference 
of the heights of the pressure 
columns. The loss of head then 
is 



//' 



Fig. 49. 



K-K, 



Art. 70.] PIEZOMETERS. 1 53 

as proved in Art. 68. This may be written 



W / 2sr ' 



If the change of section be made so that the stream has no 
subsequent enlargement, loss of head is avoided, for, as the 
above discussions show, it is the loss in velocity due to sudden 
expansion wliich causes the loss of head. 

The loss due to sudden contraction of a tube or pipe is 
usually much smaller than that due to sudden enlarge- 
ment. For instance, if the diameter of the larger section 
be three times that of the smaller, and the velocity in the 
large section be 2 feet per second, the loss of head when the 
flow passes from the smaller to the larger section is 

* = <*=^ = *ofeet 

But if the flow takes place in the opposite direction the co- 
efficient c' is about 0.64, and the loss of head is 

h' = (-4- - I)' — - = 1.6 feet, 

\O.64 / 2g 

which may be made to vanish by rounding the edges where 
the change of section occurs. 

Prob. 89. Compute the loss of head when a pipe which dis- 
charges 1.57 cubic feet per second suddenly diminishes in sec- 
tion from 12 to 6 inches diameter. 



Article 70. Piezometers. 

A piezometer is an instrument for measuring the pressure 
which exists in a pipe. In its simplest form it consists merely 



154 



FLOW THROUGH TUBES. 



[Chap. VI. 




Fig. 



of a glass tube, as aX.A, in which the water rises to a height )&, . 

At B is a form where the tube 
connecting with the pipe is of 
metal, which is joined by a flexi- 
ble hose with a glass tube, which 
may be placed alongside of a 
graduated rod to read the height 
h x . At C is a common pressure 
gauge whose dial is graduated so 
as to read either heights or pressures, as may be desired. When 
h x is found by measurement, the pressure per square unit is 
computed from the relation p 1 = wh x (Art. 9). In order to 
secure accurate results with piezometers, it is necessary that 
they be inserted into the pipe exactly at right angles ; if in- 
clined with or against the current, the height h x is greater or 
less than that due to the actual pressure at the mouth. 

If no loss of head occurs between the reservoir and the 
place where the piezometer is inserted the velocity and dis- 
charge through the pipe may be determined. The flow being 
stopped, the water in the piezometer rises to the height 1i x at 
the same level as the surface level of the reservoir ; when the 
flow occurs it stands at the height // 2 . Then 

K = K + 

whence 



v 



V = V2g(k, - /O, (46) 

and hence the discharge is known for a pipe of given size. It 
is only in cases of low velocities, however, that this method of 
gauging the flow is at all applicable, owing to the losses of 
head which always exist. 

The question as to the point from which the pressure-head 
should be measured deserves consideration. In the figures of 
the preceding articles Ji x and h n _ have been estimated upward 






Art, 70. J 



PIEZOMETERS. 



155 



= 


1^ 1 


- v= 


^3 



Fig. 50. 



from the centre of the tube, and it is now to be shown that 
this is probably correct. Let Fig. 50 represent a cross-section 
of a tube to which are attached three piezome- 
ters as shown. If there be no velocity in the 
tube or pipe, the water surface stands at the 
same level in each piezometer, and the mean 
pressure-head is certainly the distance of that 
level above the centre of the cross-section. If 
the water in the pipe be in motion, probably 
the same would hold true. Referring to formula (43) of Art. 
68, and to Fig. 46, it is also seen that if there be no velocity 
h' = h 1 — h a , which cannot be true unless h x — /z 2 = o, since 
there can be no loss of head in the transmission of static pres- 
sures ; hence h x and h^ cannot be measured from the top of the 
section. In any event, since the piezometer heights represent 
the mean pressures, it appears that they should be reckoned 
upward from the centre of the section. The absolute values 
of h x and h^ are not generally required, the difference h x — h 2 
being alone used in computations ; nevertheless the above con- 
siderations are not unimportant. 

The principal application of the piezometer is to the meas- 
urement of losses of head, as indicated in Art. 68 for the case 
of horizontal pipes. The same method applies to inclined 
pipes, only here the piezom- 
eter readings are usually 
taken above an assumed 
datum MN, as shown in 
Fig. 51. Let a x and # 2 be 
the areas of any two sec- 
tions of a pipe, v x and v 2 the 
velocities, H 1 and H the 
heights of the piezometers 
above a datum MN, and h x fig. 51. 

and h 2 the heights above the axis of the pipes, that is, the mean 




156 FLOW THROUGH TUBES. [Chap. VI. 

pressure-heads. When no flow occurs the piezometers stand 
in the same level line AB. When the flow takes place, deliv- 
ering W pounds of water per second, the effective energy in 
the first section is 

"{*■+£> 

and that in the second section is 



Now let z be the vertical distance of the centre of the second 
section below the first. Were it not for losses the energy in 
the second section w r ould be 

Wih, + Z -^) + Wz. 

Therefore the energy lost in heat due to friction, enlargement, 
contraction, and all other causes, between the two sections, is 

wl/i. + ^ + z-Zi,-^- 

V 2g 2g 

or the loss of head is 

h! = z '* ~ z '"- 4-fc+z- K. 

But from the figure it is seen that 

h x + z - K = H V —H n . 

Hence the loss of head between the two sections is 

* = *''""•' + -a; --ft. • • • • (47) 

or the same as shown in Art. 68 for horizontal tubes, the pie- 
zometer elevations being referred to the same datum. 

If the pipe be of the same diameter at the two sections the 
velocities i\ and v n _ are equal, and the loss of head is 

//' = H 1 - H, (4;)' 

which is merely the difference of level of the water surfaces 



Art. 70.] 



PIEZOMETERS. 



157 



in the piezometers. If the two sections are at the same eleva^ 
tion, or if the second section is lower than the first, this loss is 
entirely due to resistances which convert the energy into heat. 
When, however, the second section is higher than the first by 
the distance z\ the head z' is lost in overcoming the force of 
gravity, and the remainder ti —z' is the portion lost in heat. 
Piezometers therefore furnish a very convenient method of de- 
termining lost head in pipes of uniform section. For pipes of 
varying section they are rarely applied, as the discharge per 
second must be measured to find the velocities v 1 and v 2 . 

In practice it is usually the case that the piezometric tube 
is simply tapped into the top of the pipe whose flow is to be 
investigated. It is thought, however, that this may not give 
the mean pressure throughout the section. In the equations 
above deduced v 1 and v 2 are the mean velocities in the two 
sections and k x and h. 2 the corresponding mean pressure-heads. 
In order that the piezometer may correctly indicate these 
mean pressure-heads, they should perhaps be connected with 
the pipe at the sides and bottom as well as at the top. Pie- 
zometric measurements are hence liable to give results more or 
less uncertain. 

If a tube be inserted obliquely to the direction of the cur- 
rent it no longer indicates the true pressure-head, for it is 
found that the height of the water is greater when the mouth 
of the tube is inclined toward the current than when inclined 
away from it. Let 
be the angle between 
the direction of the 
flow and the inserted 
tube. Then the dy- 
namic pressure in the 
direction of the flow 




Fig. 52. 



is proportional to the velocity-head, and the component of 



I5§ FLOW THROUGH TUBES. [Chap. VI. 

this in the direction of the tube tends to increase the normal 
pressure-height h x when 6 is less than 90 and to decrease it 
when 6 is greater than 90 . Thus 

-.2 

Ji — Ji A cos 

may be written as approximately applicable to the two cases. 
In this, if the tube be inserted normal to the pipe, 6 = 90 ° and 
/i becomes /i 1 , the height due to the static pressure in the 
pipe ; if v = o, the angle 6 has no effect upon the piezometer 
readings. This discussion indicates that when the velocity v 
is great, piezometric measurements may be affected with errors 
if the connection be not made truly normal to the direction of 
the flow. 

Prob. 90. In one of the experiments on the compound tube 
shown in Fig. 53 the areas of the sections a x and a s were 
57.823 square feet, while that of a. 2 was 7.047 square feet. 
When the discharge was 54.02 cubic feet per second the pie- 
zometric elevations were : 

H i = 99.838, H, = 98.921, H 2 = 99.736 feet, 

Show that the head lost was 0.017 feet between a 1 and a 2 , and 
0.085 f eet between a 2 and a 3 . 

Article 71. The Venturt Water Meter. 

It has been shown by HERSCHEL* that a compound tube 
provided with piezometers may be used for the accurate 
measurement of water. The apparatus, which is called by him 
the VENTURI Water Meter, is shown in outline in Fig. 53, and 
consists of a compound tube (Art. 64) terminated by cylinders, 
into the top of which are tapped the piezometers H x and H % 
Surrounding the small section a. : is a chamber into which four 

* Transactions American Society of Civil Engineers, 1SS7, vol. xvii. p. 22S. 



Art. 71.] 



THE VENTURI WATER METER. 



159 



■or more holes lead from the top, bottom, and sides of the tube, 
and from which rises the piezometer H 2 . The flow passing 
through the tube has the velocities z\ , v 2 , and v 3 at the sections 
a x , a 2 , and a % , and these velocities are inversely as the areas of 
the sections (Art. 19). When the pressure in a 2 is positive the 



rr^s 




Fig. 53. 

water stands in the central piezometer at a height H 2 , as shown 
in the figure ; when the pressure is negative the air is rarefied, 
and a column of water lifted to the height h 2 . HE is the 
height of the top of the section a 2 above the datum the value 
of H 2 for the case of negative pressure was taken tobe£- h 2 . 
The apparatus was constructed so that the areas a x and a 3 were 
equal, while a 2 was about one-ninth of these. 

To determine the discharge per second through the tube, 
the areas a x and a 2 are to be accurately found by measure- 
ments of the diameters ; then 

Q =± a 1 v 1 , or Q = a 2 v 2 . 

If no losses of head occur between the sections a 1 and a 2 the 
quantity h' in the formula of the last article is o, and 

v 2 — v 2 



*g 



l6o FLOW THROUGH TUBES. [Chap. VI. 

Inserting in this for z\ and z\ their values in terms of Q, and 
then solving for Q, gives the result 

which may be called the theoretic discharge. Dividing this 
expression by a x gives the velocity v x , and dividing it by # 2 
gives the velocity v 2 . Owing to the losses of head which 
actually exist, this expression is to be multiplied by a coeffi- 
cient c\ thus: 



9 = c -^?=t' Vv( > H *- H 3 ■ ■ • (48) 

is the formula for the actual discharge per second. 

Reference is made to Herschel'S paper, above quoted, for 
a full description of the method of conducting the experi- 
ments. The discharge was actually measured either in a large 
tank or by a weir ; and thus q being known for observed pie- 
zometer heights H 1 and H if the value of c was computed by 
dividing the actual by the theoretic discharge. For example, 
the smaller tube used had the areas 

a x = 0.77288, # 2 = 0.08634 square feet; 

hence the theoretic discharge is 

2 = 0.086884 \/2g(H x — H^), 

and the coefficient of discharge or velocity is 

«-*■ 

In experiment No. 1 the value of H x was 99.069, while //„ was 
24.509 feet, and the actual discharge was 4.29 cubic feet per 
second. As E was 84.704, the value of H, is 60.195 feet. The 
theoretic discharge then is 



Q = 0.086884 X 8.02 V 38.874 = 4.345. 



Akt. 7i.] THE VENTURI WATER METER. l6l 

Dividing 4.29 by this, gives for c the value 0.988. Fifty-five 
experiments made in this manner, in all of which negative 
pressure existed in a 2 , gave coefficients ranging in value from 
0.94 to 1.04, only four being greater than 1.01 and only two 
less than 0.96. 

The larger tube used had the areas a 1 = 57.823 and 
# 2 = 7.074 square feet, and the pressure at the central piezom- 
eter was both positive and negative. Twenty-eight experi- 
ments give coefficients ranging from 0.95 to 0.99, the highest 
coefficients being for the lowest velocities. In this tube the 
velocity at the section a 2 ranged from 5 to 34.5 feet per second. 
The small variation in the coefficients for the large range in 
velocity indicates that the apparatus may in the future take a 
high rank as an accurate instrument for the measurement of 
water. Under low velocities, however, it is not probable that 
the arrangement of piezometers shown in Fig. 53 will give the 
best results ; in order that H 1 may correctly indicate the mean 
pressure in a x , connection seems to be required both at the 
bottom and sides of the tube like that at # 3 . It is thought, 
moreover, that the elevation E should be measured to the 
centre of the section rather than to the top. The lower pie- 
zometer H z is not an essential part of the apparatus and may 
be omitted, although it was of value in the experiments as show- 
ing the total loss of head. 

Prob. 91. Given # 2 = 7.074 and a l = 57.823 square feet, 
h 2 = 12.204, E — 90.909, and H x = 98.773 feet, to compute 
the coefficient of discharge when q = 243.87 cubic feet. 



*62 FLOW THROUGH PIPES. [Chap, Mil 



CHAPTER VII. 
FLOW THROUGH PIPES. 

Article 72. Fundamental Ideas. 

The simplest case of flow through a pipe is that where the 
discharge occurs entirely at the end, there filling the entire sec- 
tion, as in a tube ; such pipes are said to be in a condition of 
full flow. Other cases are those where the discharge is drawn 
from the pipe at several points along its length, as in the water 
mains for the supply of towns. Pipes with full flow will be 
first considered, but most of the principles and tables relating 
to them apply with but slight modification to water mains. 
Pipes used in engineering practice are rarely less than |- inch 
in interior diameter, and may range from this value upward to 
4 feet or more. 

The phenomena in a pipe with full flow are apparently sim- 
ple. The water from the reservoir, as it enters the pipe, suffers 

more or less contraction, depend- 
ing upon the manner of connec- 
^ tion, as in tubes. Its velocity is 
K<-Z then retarded by the resistances 
of friction and cohesion along the 
interior surface, so that the dis- 
IG ' 54 ' charge at the end is much smaller 

than in the tube. When the flow becomes permanent the pipe 
is entirely filled throughout its length ; and hence the mean 
velocity at any section is the same as that at the end, if the 
size be uniform. This velocity is found to decrease as the 




Art. 72.] FUNDAMENTAL IDEAS. Il6$ 

length of the pipe increases, other things being equal, and be- 
comes very small for great lengths, which shows that nearly all 
the head has been lost in overcoming the resistances. 

The head which causes the flow is the difference in level 
from the surface of the water in the reservoir to the centre of 
the end, when the discharge occurs freely into the air. If h be 
this head, and W the weight of water discharged per second, 
the theoretic energy per second is Wh ; and if v be the actual 

velocity or discharge the effective energy is . The lost 



energy is then W\k — — , and this has disappeared in heat in 



overcoming the resistances. In other words, the total head is 

h, the effective head of the outflowing stream is - — , and the 

2g 

lost head is h — — . If the lower end of the pipe is sub- 

merged, as is often the case, the head h is the difference in 
elevation between the two water levels. 

The length of a pipe is measured along its axis, following all 
its windings if any. When the length is about two and one- 
half diameters the pipe is a tube whose coefficient of discharge 
varies from 0.71 to 0.82, according to the arrangement of its 
inner end (Art. 65). As the length increases the coefficient of 
discharge becomes less than from the tube, and for long pipes 
it becomes very small indeed — indicating that the greater part 
of the head h is expended in heat in overcoming resistances. 

The object of the discussion of flow in pipes is to enable the 
discharge which will occur under given conditions to be deter- 
mined, or to ascertain the proper size which a pipe should 
have in order to deliver a given discharge. The subject can- 
not, however, be developed with the definiteness which char- 
acterizes the flow from orifices and weirs, partly because the 



164 FLOW THROUGH PIPES. [Chap. VIL 

condition of the interior surface of the pipe greatly modifies 
the discharge, partly because of the lack of experimental data, 
and partly on account of defective theoretical knowledge re- 
garding the laws of flow. In orifices and weirs errors of two 
or three per cent may be regarded as large with careful work ; 
in pipes such errors are common, and are generally exceeded 
in most practical investigations. It fortunately happens, how- 
ever, that in most cases of the design of systems of pipes errors 
of five and ten per cent are not important, although they are 
of course to be avoided if possible, or, if not avoided, they 
should occur on the side of safety. 

Prob. 92. A pipe 500 feet long and 3 inches in diameter dis- 
charges about 48 gallons per minute under a head of 4 feet. 
Compute the coefficient of discharge. 

Article 73. Loss of Head at Entrance. 

The loss of head which occurs in the upper end of the pipe, 
due to contraction and resistance of the inner edges, is called 
the loss at entrance, and this is the same as in a short cylin- 
drical tube under the same velocity of flow. Let c be the 
coefficient of velocity or discharge for a short tube and v the 
mean velocity at its outer end, then (Art. 66) the loss of head 
in the tube is 

V / 2g 

Now this velocity v is the same as that in the pipe into which 
the tube may be regarded as discharging, and hence this same 
expression is the loss of head which occurs at the entrance of 
the pipe, or rather it is the loss at the upper end in a length 
equal to about three diameters. 

The discussions of the last chapter show that the mean 
value of c is about 0.72 when the tube projects into the reser- 
voir, about 0.82 when the inner end is flush with side of the 



Art. 73.] LOSS OF HEAD AT ENTRANCE. 165 

reservoir and has square corners, and that it may be nearly 1.00 
when the inner end is provided with a bell-shaped mouth. 
Accordingly the loss of head for a pipe projecting into the 
reservoir is 



*g 



h! — ( i — 1 ) — = 0.93 

\O.72 / 2g 

and for a pipe whose end is arranged like a standard tube, 

h' = I— 5-1 — 1 — = 0.49 — ; 

\O.82 2 / 2g y 2g 

and for a pipe with a perfect mouthpiece, 



// =l? -,]- = o. 



The loss of head at entrance is hence always less than the 
velocity-head, and it may be expressed by the formula 

h '^ m 2~g' (49) 

in which m is 0.93 for the inward projecting pipe, 0.49 for the 
standard end, and o for a perfect mouthpiece. When the con- 
dition of the end is not specified the value used for m in the 
following pages will be 0.5, which supposes that the arrange- 
ment is like the standard tube or nearly so. For short pipes, 
however, it may be necessary to consider the particular condi- 
tion of the end, and then 

>*= ( ? - ij, ..... • (49V 

in which c is to be selected from the evidence presented in the 
last chapter. 

It should be noted that the loss of head at entrance is very 
small for long pipes. For example it is proved by actual 
gaugings that a pipe 10 000 feet long and 1 foot in diameter 
discharges about 4J cubic feet per second under a head of 100 
feet. The mean velocity then is 
4.25 



0.7854 



= 5.41 feet per second, 



1 66 FLOW THROUGH PIPES. [Chap. VII. 

and the probable loss of head at entrance hence is 

k! — 0.5 X 0.01555 X 541' 2 = 0.228 feet, 
or only one-fourth of one per cent of the total head. In this 
case the effective velocity-head of the issuing stream is only 
0.455 f eet > which shows that the total loss of head is 99.545 
feet. 

Prob. 93. Under a head of 20 feet a pipe 1 inch in diameter 
and 100 feet long discharges 15 gallons per minute. Compute 
the loss of head at entrance. 

Article 74. Loss of Head in Friction. 

The loss of head due to the resisting friction of the interior 
surface of a pipe is usually large, and in long' pipes it becomes 
very great, so that the discharge is but a small percentage of 
that due to the head. Let h be the total head on a pipe with 

full flow, — the velocity-head of the issuing stream, h' the head 

lost at entrance, and h" the head lost in frictional resistances. 
Then if the pipe be straight and of uniform size, so that no 
other losses occur, 2 

k = — +k'-\- h". 

Inserting for h! its value from Art. 73, this equation becomes 

2£* V / 2? 

which is a fundamental formula for the discussion of flow in 
pipes. 

The head lost in friction may be determined for particular 
cases by measuring the head /i, the area a of the cross-section 
of the pipe, and the discharge per second q. Then q divided 
by a gives the mean velocity v, and from the above equation 

h" =A--,*—, 

c 22? 



Art. 74.] LOSS OF HEAD IN FRICTION. 1 67 

which serves to compute h'\ the value of c being first selected 
according to the condition of the end. This method is not 
applicable to very short pipes because of the uncertainty 
regarding the coefficient c (Art. 65). 

Another method, and the one most generally employed, is 
by the use of piezometers (Art. 70). A portion of the pipe 
being selected which is free from sharp curves, two vertical 
tubes are inserted into which the water rises. The differ- 
ence of level of the water surfaces in the piezometers is then 
the head lost in the pipe between them, and this loss is caused 
by friction alone if the pipe be straight and of uniform size. 

By these methods many experiments have been made upon 
pipes of different sizes and lengths under different velocities of 
flow, and the discussion of these has enabled the approximate 
laws to be deduced which govern the loss of head in friction, 
and tables to be prepared for practical use. These laws are : 

1. The loss in friction is proportional to the length of the 

pipe. 

2. It increases nearly as the square of the velocity. 

3. It decreases as the diameter of the pipe increases. 

4. It increases with the roughness of the interior surface. 

5. It is independent of the pressure of the water. 

These laws may be expressed by the equation 

k"=*/if (50) 

in which / is th^e length of the pipe, d its diameter, and f is a 
quantity which depends upon the degree of roughness of the 
surface. This equation is an empirical one merely ; the theo- 
retic expression for h" is as yet unknown, and it is probable 
that when discovered it will prove to be of a complex nature. 

The values of h" having been deduced for a number of 



1 68 



FLOW THROUGH PIPES. 



[Chap. VII. 



cases in the manner just explained, the corresponding values 
of /can be computed. In this manner it is found that /varies 
not only with the roughness of the interior surface of the pipe, 
but also with its diameter, and with the velocity of flow. From 
the discussions of Fanning, Smith, and others, the following 
table of mean values of / has been compiled, which are appli- 
cable to clean iron pipes, either smooth or coated with coal- 
tar varnish, and laid with close joints. 

TABLE XVI. FRICTION FACTORS FOR PIPES. 



Diameter 






Velocity 


n Feet per 


Second. 






in 
Feet. 


1. 


2. 


3- 


«■ 


6. 


10. 


15. 


O.05 


0.047 


0.041 


O.037 


O.034 


O.031 


O.O29 


0.028 


O. I 


.038 


.032 


.030 


.028 


.026 


.024 


.023 


O.25 


.032 


.028 


.026 


.025 


.024 


.022 


.021 


0-5 


.028 


.026 


• 025 


.023 


.022 


.020 


.OI9 


0-75 


.026 


.025 


.024 


.022 


.021 


.OI9 


.018 


I. 


.025 


.024 


.023 


.022 


.020 


.Ol8 


.OI7 


I.25 


.024 


.023 


.022 


.021 


.OI9 


.OI7 


.016 


1-5 


.023 


.022 


.021 


.020 


.018 


.Ol6 


.OI5 


i-75 


.022 


.021 


020 


• Ol8 


.OI7 


.015 


.OI4 


2. 


.021 


.020 


.OI9 


.017 


.016 


.014 


.013 


2-5 


.020 


.019 


.Ol8 


.Ol6 


.OI5 


.013 


.OI2 


3- 


.019 


.018 


.Ol6 


.OI5 


.OI4 


.013 


.012 


3-5 


.018 


.017 


.Ol6 


.014 


.013 


.OI2 




4- 


.017 


.016 


.OI5 


.013 


.012 


.Oil 




5- 


.016 


.015 


.OI4 


.OI3 


.012 






6. 


.015 


.014 


.OI3 


.012 


.Oil 







The quantity /may be called the friction factor, and the 
table shows that its value ranges from 0.05 to 0.01 for new 
clean pipes. A rough mean value, often used in approximate 
computations, is 

Friction factor /= 0.02. 



Art. 74] LOSS OF HEAD IN FRICTION. 1 69 

It is seen that the tabular values of /decrease both when the 
diameter and when the velocity increases, and that they vary 
most rapidly for small pipes and low velocities. The probable 
error of a tabular value of /is liable to be about one unit in 
the third decimal place, which is equivalent to an uncertainty 
of ten per cent when f = 0.011, and to five per cent when 
f =1 0.021. The effect of this is to render computed values of 
h" liable to the same uncertainties; but the effect upon com- 
puted velocities and discharges is much less, as will be seen 
in Art. 76. 

To determine, therefore, the probable loss of head in fric- 
tion, the velocity v must be known, and f'\s taken from the 
table for the given diameter of pipes. The formula 

d 2g 

then gives the probable loss of head in friction. For example, 
let / = 10 000 feet, d= 1 foot, v = 5.41 feet. Then, from the 
table, /"is 0.021, and 

h" = 0.021 X 1 ° i ° ° X 0.455 = 96 feet, 
which is to be regarded as an approximate value, liable to an 
uncertainty of five per cent. 

The theory of the internal frictional resistances, as far as 
understood, indicates that the energy which is thus transformed 
into heat is expended in two ways: first, in the direct friction 
along the interior surface ; and second, in impact caused by an 
unsteady motion of the particles of water. Under very low 
velocities the motion is in lines parallel to the axis of the pipe, 
so that resistance is met only along the surface, but under ordi- 
nary conditions the motion of many of the particles is sinuous, 
whereby internal friction or impact is also produced. Experi- 
ments devised by REYNOLDS enable this sinuous motion to be 
actually seen, so that its existence is beyond question. 



I70 FLOW THROUGH PIPES. [Chap. VII. 

Prob. 94. Determine the actual loss of head in friction from 
the following experiment : / = 60 feet, h — • 8.33 feet, d = 
0.0878 feet, q = 0.03224 cubic feet per second, and c = 0.8. 
Compute by help of the table the probable loss for the same 
data. 

Article 75. Other Losses of Head. 

Thus far the pipe has been supposed to be straight and o\ 
uniform size, so that no losses of head occur except at en- 
trance and in friction. But if the pipe vary in diameter, or 
have sharp curves, or contain valves, further losses occur, which 
are now to be considered. 

Sudden enlargements and contractions of section cause 
losses of head which may be ascertained by the rules of Arts. 
68 and 69. These are of infrequent occurrence in pipes, the 
usual method of passing from one size to another being by 
means of a " reducer," which is a conical frustum several feet 
long, whereby the velocity is slowly changed without expend- 
ing energy in impact. 

The loss of head caused by easy curves is very slight, and 
need not be taken into account. For sharp curves the loss is 
small, rarely exceeding twice the velocity-head for a single 
curve, but when many such curves occur the item of loss thus 
caused may be important. According to the investigations of 
WEISBACH, the loss of head due to a curve of one-fourth of a 
circle may be written 

h m — n — , 

in which n is a number whose value is given below for different 

d 

values of — =. where R is the radius of the curve of the centre 
2R 

line of the pipe, and d is its diameter : 

-g .= 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 
n — 0.13, 0.14, 0.16, 0.21, 0.29, 0.44, 0.66, 0.98, 1. 41, 1.98 



Art. 75.] OTHER LOSSES OF HEAD. 171 

These coefficients, however, were derived for small pipes, and 
it is probable that for large pipes the loss of head may be less 
than they indicate. 

In Fig. .55 are shown three kinds of valves for regulating 
the flow in pipes : at A a valve consisting of a vertical sliding- 
gate, at B a cock-valve formed by two rotating segments, and 



zzzzzzzzzzzzzzzzzza 




Fig. 55. 

at C a throttle-valve or circular disk which moves like a damper 
in a stove-pipe. The loss of head due to these may be very 
large when they are sufficiently closed so as to cause a sudden 
change in velocity. It may be expressed by 

h!" = n — , 

in which n has the following values, as determined by the ex- 
periments of WEISBACH.* For the sluice-valve let d' be the 
vertical distance that the gate is lowered below the top of the 
pipe ; then 



"For — n 1 X 3 15 37 

-TOr— — O 8 ¥ "8 2 -g- ¥ ¥ 



d 

n — 0.0 0.07 0.26 0.81 2.1 5.5 17 98 



For the cock-valve let 6 be the angle through which it is 
turned, as shown in the figure ; then 

For 6 = o° io° 20 30 40 50 55 6o° 65 

n — o 0.29 1.6 5.5 17 53 106 206 486 

* Mechanics of Engineering, vol. i., Coxe's translation, p. 902. 



I7 2 FLOW THROUGH PIPES. [Chap. VII. 

In like manner, for the throttle-valve the coefficients are : 

For 0= 5 io° 20° 30 40 50 6o° 65 ;o° 

72 = 0.24 0.52 1.5 3.9 11 3.3 118 256 750 

The number n hence rapidly increases and becomes infinity 
when the valve is fully closed, but as the velocity is then zero 
there is no loss of head. The velocity v here, as in other 
cases, refers to that in the main part of the pipe, and not to 
that in the contracted section formed by the valve. 

An accidental obstruction in a pipe may be regarded as 
causing a sudden change of section, and the loss of head due 
to it is, by Art. 68, 



fe - l ) ' 



V 

~*g 



where a is the area of the section of the pipe, and a' that of 
the diminished section. This formula shows that when a' is 
one-half of a y the loss of head is equal to the velocity-head, 
and that n rapidly increases as a' diminishes. 

In the following pages the symbol 1i" will be used to 
denote the sum of all the losses of head due to curvature, 
valves, and contractions of section. Then 

h" f =n^, (51 

in which n will denote the sum of all the coefficients due to 
these causes. In case no mention is made regarding these 
sources of loss they are supposed not to exist, so that both n 
and ti" are simply zero. 

Prob. 95. Compute for the data of the last problem the 
loss of head caused by a semi-circular curve whose radius is 2 
inches. 



Art. 76.] FORMULA FOR VELOCITY. 173 

Article 76. Formula for Velocity. 

The mean velocity in a pipe can now be deduced for the 
condition of full flow. The total head being k, and the effec- 

tive velocity-head of the issuing stream being — , the lost head 

<5 

is h , and this must be equal to the sum of its parts, or 

k-?- = k> + h"+k>" (52) 

Substituting in this the values of //, k"., and h'" from the pre- 
ceding articles, it becomes 



/ v 2 
f- 

and by solving for v there is found 



h — m \-f-j \- n — ; . . (52)' 



I 2gh 



I 



(53) 



d 

which is a general formula for the velocity of flow. 

In this formula n will be taken as o, unless otherwise stated ; 
that is, no losses of head occur except at entrance and in fric- 
tion. The formula for pipes which are essentially straight and 
of uniform size throughout then is 



v = 




(53)' 



Here m is taken as 0.5, which is to be regarded as its mean 
value in accordance with the discussion in Art. 73. 

In this formula the friction factor /"is a function of v to be 
taken from the table in Art. 74, and hence v cannot be directly 



174 FLOW THROUGH PIPES. [Chap. VII. 

computed, but must be obtained by successive approximations. 
For example, let it be required to compute the velocity of dis- 
charge from a pipe 3000 feet long and 6 inches in diameter 
under a head of 9 feet. Here / = 3000, ^/ = 0.5, and h = 9 ; 
taking for /the rough mean value 0.02, the formula gives 



2 X 32.16 X 9 

v = \ I ; = 2.2. 

1.5 + 0.02 X 3000 X 2 

The approximate velocity is hence 2.2 feet per second, and 
entering the table with this, the value of /is found to be 0.026. 
Then the formula gives 



/ 2 X 3 2 -!6 X 9 



-7- = I.Q2. 

-f- 0.026 X 3000 X 2 ^ 



This is to be regarded as the probable value of the velocity, 
since the table gives / = 0.026 for v = 1.92. In this manner 
by one or two trials the value of v can be computed so as to 
agree with the corresponding value of/. 

The error in the computed velocity due to an error of one 
unit in the last decimal of the factor / is always relatively less 
than the error in /itself. For instance, if v be computed for 
the above example with/= 0.025, its value is found to be 1.96 
feet per second, or two per cent greater than 1.92. In general, 
the percentage of error in v is less than one-half of that in / 
It hence appears that computed velocities are liable to probable 
errors ranging from one to five per cent, owing to imperfections 
in the tabular values of / for new clean pipes. This uncer- 
tainty is as a rule still further increased by various causes, so 
that five per cent is to be regarded as a common probable error 
in computations of velocity and discharge from pipes. 

Velocities greater than 15 feet per second are very unusual 
in pipes, and but little is known as to the values of/ for such 
cases. For velocities less than one foot per second, the values 



Art. 77-] COMPUTATION OF DISCHARGE. 1 75 

of /are also not understood, so that little reliance can be placed 
upon computations. The usual velocity in water mains is less 
than five feet per second, it being found inadvisable to allow 
swifter flow on account of the great loss of head in friction. 

To illustrate the use of the general formula, let the pipe in 
the above example be supposed to have a curve of 6 inches 
radius, and to contain a gate valve which is half closed. Then 
from Art. 75, n = 0.29 for the curve and n = 2.1 for the valve, 
or in the formula n is to be put as 2.39. The velocity is now 
found to be 



2 X 32.16 X 9 r - , 

v = \ / —z — . 7 7 = 1. QO teet per second ; 

Y 3.89 + 0.026 X 6000 y F 

which is but a trifle less than that found before. The closing- 
of the sluice gate to one-half its depth hence but slightly in- 
fluences the velocity, while the effect of the curve is scarcely 
perceptible. With a shorter pipe, however, the influence of 
these would be more marked. 

Prob. 96. Compute the velocity for the data of the last 
example if the pipe be 1000 feet long. 

Prob. 97. Compute the velocity for a pipe 15 000 feet long 
and 18 inches in diameter under a head of 230 feet. 

Ans. 9.57 feet per second. 

Article yy. Computation of Discharge. 

The discharge per second from a pipe of given diameter is 
found by multiplying the velocity of discharge by the area of 
the cross-section of the pipe, or 

g = l7td 2 V = 0.7854^, (54) 

in which v is to be found by the method of the last article. 

For example, let it be required to find the discharge in 
gallons per minute from a clean pipe 3 inches in diameter and 



176 FLOW THROUGH PIPES. [Chap. VIL 

500 feet long under a head of 4 feet. Here d = 0.25, /= 500, 
and h = 4. Then ior/= 0.02, the velocity is found to be 2.5 
feet per second ; again taking from the table f = 0.027, the 
velocity is 2.15 feet per second. The discharge in cubic feet 
per second is 

q = O.7854 X O.25 2 X 2.15 — 0.106; 

and in gallons per minute, 

q = O.106 X 7.48 X 60 = 47.6. 

This is the probable result, which is liable to the same uncer- 
tainty as the velocity — say about three per cent ; so that strictly 
the discharge should be written 47.6 ± 1.4 gallons. 

By inserting the value of v in the above expression for q it 
becomes 



q =w 



*+™+4 



and from this the value of the head required to produce a 
given discharge is 

T^ / 



' 2g7t i V l ' J d ' Id*' 

These formulas are not more convenient for practical computa- 
tions than the separate expressions for z>, q, and // previously 
established, since in any event v must be computed in order to 
select f from the table. They serve, however, to exhibit the 
general laws which govern the discharge. 

Prob. 98. Compute the probable discharge from a pipe 1 
inch in diameter and 1000 feet long under a head of 40 feet. 

Prob. 99. What head is required to discharge 3 gallons per 
minute through a pipe 1 inch in diameter and 1000 feet long? 

Ans. 1 1.3 feet. 



Art. 78.] COMPUTATION OF DIAMETER. 1 77 

Article 78. Computation of Diameter. 

It is an important practical problem to determine the 
diameter of a pipe to discharge a given quantity of water 
under a given head and length. The last equation above 
serves to solve this case, as all the quantities in it except d 
are known. This may be written in the form 

,. = [ (I +.«+-. y + //]J!£ ; 

or placing for m and 2g their mean values and neglecting n, it 
becomes 

~" /7 2 

^=0.479^(1.5^ + //)! 



(55) 



which is the formula for computing d when h, /, and d are in 
feet and q is in cubic feet per second. The value of the fric- 
tion factor /may be taken as 0.02 in the first instance, and the 
d in the right-hand member being neglected, an approximate 
value of the diameter is computed. The velocity is next 
found by the formula 

\nd* 

and from Table XVI. the value of f thereto corresponding is 
selected. The computation for d is then repeated, placing in 
the right-hand member the approximate value of d. Thus 
by one or two trials the diameter is computed which will 
satisfy the given conditions. 

For example, let it be required to determine the diameter 
of a pipe which, under the condition of full flow, will deliver 
500 gallons per second, its length being 4500 feet and the 
head 24 feet. Here the value of q is 

500 
q — —77- = 66.84 cubic feet. 
1 748i 



1/3 FLOW THROUGH PIPES. [Chap. VII. 

The approximate value of d then is 

/0.02 x 4500 x 66.84 2 U 
d = o.479( ^ - f == 3-35 feet. 

From this the velocity of flow is 

66.84 

z> = — a — i = 7.6 feet per second, 

0.7854 X 3-35 F 

and from the table the value of /for this diameter and velocity 
is found to be 0.013. Then 

66.84 s 



d = 0.479 



(i-5 x 3.35+0.013 x 4500) 



24 

from which d — 3.125 feet. With this value of d the velocity 
is now found to be 8.71 feet, so that no change results in the 
value of f. The required diameter of the pipe is therefore 
3.1 feet, or about 37 inches ; but as the regular market sizes of 
pipes furnish only 36 inches and 40 inches, one of these must 
be used, and it will be on the side of safety to select the 
larger. 

It will be well in determining the size of a pipe to also con- 
sider that the interior surface may become rough by erosion 
and incrustation, thus increasing the value of the friction fac- 
tor and diminishing the discharge. The increase in f from 
these causes is not likely to be so great in a large pipe as in a 
small one, but it is thought that for the above example they 
might be sufficient to make / as large as 0.03. Applying this 
value to the computation of the diameter from the given data 
there is found d = 3.6 feet = about 43 inches. 

The sizes of pipes generally found in the market are 4-, f , 
1, ii if, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 27, 30, 36, 40, 44, 
and 48 inches, while intermediate or larger sizes must be made 
to order. • The computation of the diameter is merely a guide 
to enable one of these sizes to be selected, and therefore it 



Art. 79.] SHORT PIPES. 1 79 

is entirely unnecessary that the numerical work should be car- 
ried to a high degree of precision. In fact, three-figure loga- 
rithms are usually sufficient to determine reliable values of d. 

Prob. 100. Compute the diameter of a pipe to deliver 50 
gallons per minute under a head of 4 feet when its length is 
500 feet. Also when its length is 5000 feet. 

Article 79. Short Pipes. 

A pipe is said to be short when its length is less than about 
500 times its diameter, and very short when the length is less 
than about 50 diameters. In both cases the coefficient c should 
be estimated according to the condition of the upper end as 
precisely as possible, and the length / should not include the 
first three diameters of the pipe, as that portion properly be- 
longs to the tube which is regarded as discharging into the 
pipe. In attempting to compute the discharge for such pipes, 
it is often found that the velocity is greater than given in 
Table XVI., and hence that the friction factor f cannot be de- 
termined. For this reason no accurate estimate can be made 
of the discharge from short pipes under high heads, and fortu- 
nately it is not often necessary to use them in engineering 
constructions. 

For example, let it be required to compute the velocity of 
flow from a pipe 1 foot in diameter and 100 feet long under a 
head of 100 feet, the upper end being so arranged that c = 0.80, 
and hence 771=0.56 (Art. 73). Neglecting 77, since the pipe 
has no curves or valves, the formula for the velocity becomes 

and using for f the rough mean value 0.02, 



v = 



(..32 X IOO 



1.56 -j- 0.02 x 97 



42.9 feet per second. 



I So FLOW THROUGH PIPES. [Chap. VII. 

Now there is absolutely no experimental knowledge regarding- 
the value of the friction factor f for such high velocities, but 
judging from the table it is probable that / may be about 
0.015. Using this instead of 0.02 gives for v the value 46 feet 
per second. The uncertainty of this result should be regarded 
as at least ten per cent. 

For very short pipes there are on record a number of ex- 
periments by EYTELWEIN and others, from which the coeffi- 
cients of discharge have been deduced. The upper end of the 
pipe being in all cases arranged like the standard tube, these 

experiments give the following as mean values of the velocity : 

For / = id. v — 0.S2 \ 2gh 

For I = 12*/, v = 0.77 \ 2gh 

For / = 244 v = 0.73 \ 2gh 

For / = $6d, v = 0.6S S ~2gh 

For / = 4&/, v — 0.63 \ 2gh 

For / = 60^/, v = 0.60 S' 2gh 

These coefficients were deduced for small pipes under low 
heads, and are to be regarded as liable to a variation of severaL 
per cent : for large pipes and high heads they are all probably 

too large. 

The general equation for the velocity of discharge deduced 
in Art. j6 may be applied to very short pipes by writing / — 3d 
in place of /. and placing for ;;/ its value in terms of c. It then 
becomes 



v 



2gh 


I 

c 




-Id 
d 



(56) 



If in this / equals id, the velocity is 

V = C\ 2gh> 



Art. 80.] LONG PIPES. l8l 

which is the same as for a short tube. If / = \2d, f = 0.02, 
and c = 0.82, it gives v — 0.774 V2gh, which agrees well with 
the mean value above stated. 

Prob. 101. Compute the discharge per second for a pipe 
1 inch in diameter and 40 inches long under a head of 4 feet. 

Article 80. Long Pipes. 

For long pipes the loss of head at entrance becomes very- 
small compared with that lost in friction, and the velocity-head 
is also small. The formula for velocity deduced in Art. j6 is 

/ 2?h 



V-'-s+fi 



d' 

in which the first term in the denominator represents the effect 
of the velocity-head and the entrance-head, the mean value of 
the latter being 0.5. Now it may safely be assumed that 1.5 
may be neglected in comparison with the other term, when the 
error thus produced in v is less than one per cent. Taking for 
f its mean value, this will be the case when 



/ 



1. 01, from which -= = 3750. 



1/ 



7 ' d 



0.02 —, 
a 



Therefore when / is greater than about 400CV/ the pipe will 
be called long. 

For long pipes the velocity under full flow hence is given 
by the formula 

v = \^ = ,,Wf r .... ( 57 ) 

and the discharge per second is, 

* = WV = 6.3o*/^j (57)' 



1 82 FLOW THROUGH PIPES. [Chap. VII. 

For computing the diameter required to deliver a given dis- 
charge the formula is 

d=o A79 ff) ........ (5 7 )- 

These equations show that for very long pipes the discharge 
varies directly as the 2\ power of the diameter, and inversely 

as the square root of the length. 

In the above formulas d, //, and / are to be taken in feet, q in 
cubic feet per second, and f is to be found from the table in 
Art. 74, an approximate value of v being first obtained by 
taking /"as 0.02. It should not be forgotten that these expres- 
sions are of an empirical nature, and do not necessarily rep- 
resent the true laws of flow ; but at present they seem to be the 
representation of these laws which for long pipes best agrees 
with experiments. The value of // in these formulas is also 
really the friction-head h" , since in their deduction the other 

heads, ti , h"' and — , have been neglected ; these, however, al- 

though often very small, can never be really zero. 

Prob. 102. Compute the probable discharge from a pipe 
26 500 feet long and 18 inches in diameter under a head of 
324.7 feet. 

Prob. 103. Compute the diameter required to deliver 15 000 
cubic feet per hour through a pipe 26 500 feet long under a 
head of 324.7 feet. If this quantity is carried in two pipes of 
equal diameter, what should be their size? 

Article 81. Relative Discharging Capacities. 

For orifices and short tubes the discharge under a given 
head varies as the square of the diameter. In pipes of equal 
length under given heads the discharges vary mere rapidly 



Art. 8i.] RELATIVE DISCHARGING CAPACITIES. 1 83 

than the squares of the diameters, owing to the influence of fric- 
tion. For a long pipe the- formula for discharge is 



2ghd b 

which shows that if f be constant the discharge varies as the 
2j power of the diameter. This is a useful approximate rule 
for comparing the relative discharging capacities of pipes. 

Thus if there be two pipes with diameters d 1 and d^ the rule 
gives 

q x :q, = df-di\ ...... (58) 

and from this 

I)'. ..... . (5 sy 

For example, if there be two pipes of the same length under 
the same head, the first one foot and the second two feet in 
diameter, 

& = q, (y)* = 57&> 

or the second pipe discharges nearly six times as much as the 
first. In other words, six pipes of 1 foot diameter are about 
equivalent to one pipe of 2 feet diameter. 

As the friction factor /"is not constant, the above rule is not 
exact ; for, as the formula shows, 

from which 



'• ■ '• = w ■ W • 



Now as the values of f vary not only with the diameter but 
with the velocity, a solution, cannot be made except in partic- 
ular cases. For the above example let the velocity be about 



1 84 FLOW THROUGH PIPES. [Chap. VII. 

3 feet per second ; then from the table /j = 0.023 and 
f % — 0.019, and 

q, = q* (2) f (i.2)^ = 6.2^, 

or the two-foot pipe discharges more than six times as much 
as the one-foot pipe. 

Prob. 104. How many pipes, 6 inches in diameter, are equiv- 
alent in discharging capacity to one pipe 24 inches in diameter? 



Article 82. A Compound Pipe. 

A compound pipe is one laid with different sizes in differ- 
ent portions of its length. In such the change from one size 
to another is to be made gradually by a reducer, so that losses 
of head due to sudden enlargement or contraction are avoided 
(Art. 68). Let d lf d 2 , d 3 , etc., be the diameters ; l x , / 3 , / 3 , etc., 
the corresponding lengths, the total length being l x -\- / 3 -f- etc. 
Let z\, v 2 , etc., be the velocities in the different sections. 
Neglecting the loss of head at entrance, the total head h may 
be placed equal to the loss of head in friction, or 

/z=/i iS +/ ^S +etc -- • • (6o) 

Now if the discharge per second be q, 

v = M. v = **- etc 

1 red?' 2 ndr 

Substituting these and solving for q, gives 



-, — 2 - gh , , .... (60)' 

/,4+/A+etc. 



in which / lt / 2 , etc., are the friction factors corresponding to 
the given diameters and velocities in Table XVI. 



Art. 82.] A COMPOUND PIPE. 1 85 

For example, take the case of two sizes for which the 
dimensions are 

d x = 2 feet, l x — 2800 feet, 

d t = 1.5 feet, / 2 == 2145 f eet > ^ = I2 7-5 f eet - 

Using for/j and f 9 the mean value 0.02, and making the sub- 
stitutions in the formula, there is found 

q = 27.3 cubic feet per second, 

from which z\ = 8.7 and v 2 = 15.4 feet per second. 

Now from the table in Art. 74 it is seen that f x = 0.015 and 
f 3 z= 0.015 ; and repeating the computation, 

q = 30.1 cubic feet per second, 

which gives z\ = 9.60 and v 2 = 17.0 feet per second. 

These results are probably as definite as the table of friction 
factors will allow, but are to be regarded as liable to an uncer- 
tainty of several per cent. 

To determine the diameter of a pipe which will give the 

same discharge as the compound one, it is only necessary to 

I 
replace the denominator in the above value of q by f~ji, where 

/ = /j -J- / a -\- etc., and d is the diameter required. Taking the 
values of /as equal, this gives 

— = — - + — + etc. 

d b d; ^ d: ^ 

Applying this to the above example, it becomes 

4945 = 2800 , 2145 
d*~ 2 b ■ i.5 5 ' 

from which d = 1.68 feet, or about 20 inches. 



1 86 



FLOW THROUGH PIPES. 



[Chap. VII. 



Compound pipes are sometimes used in order to prevent 
the hydraulic gradient (Article 84) from falling below the pipe. 
At Rochester, N. Y., there is a pipe 101 261 feet long, of which 
50 7~6 feet is 36 inches in diameter, and 50 485 feet is 24 inches 
in diameter. Under a head of 385.6 feet this pipe discharged 
in 1876 about 14 cubic feet per second and in 1890 about 10J- 
cubic feet per second. 

Prob. 105. Compute the discharge of the Rochester pipe, 
using the table on page 168. 

Article 83. Piezometer Measurements. 



Let a piezometer tube be inserted into 

point Z> 




a pipe at any 
whose distance 
from the reservoir is l x 
measured along the pipe 
line. Let A 1 D 1 be the 
vertical depth of this 
point below the water 
level of the reservoir : 
FlG - 56 ' then if the flow be stopped 

at the end C, the water rises in the tube to the point A y . But 
when the flow occurs, the water level in the piezometer stands 
at some point C i , and the piezometric height or pressure-head 
is h- L , or C 1 D l in the figure. The distance A X C } then represents 
the velocity-head plus all the losses of head between Z\ and 
the reservoir. If no losses of head occur except at entrance 
and in friction, the value of A 1 C 1 then is 



^ = fr + 



m 



f 



d 2, 



(61) 



from which the piezometric height can be found when v has 
been determined by measurement or by computation. 

For example, let the total length / = 3000 feet. cf= 6 inches, 
h — 9 feet, and ;;/ = 0.5. Then, as in Art. y6 % there is found 



Art. S3.] PIEZOMETER MEASUREMENTS. 1 87 

/;-=. 0.026 and v = 1.917 feet per second. The position of 
the top of the piezometric column is then given by 

H x = (1.5 +0.052/O x 0.05714, 

and the height of that column is 

h x = A X D X - H x . 

Thus if l x — 1000 feet, H x = 3.06 feet ; and if l x = 2000 feet, 
H x = 6.03 feet. If the pipe is so laid that A X D X is 9 feet, the 
corresponding pressure-heights are then 5.94 and 2.97 feet. 

For a second piezometer inserted at D 2 at the distance / 2 
from the entrance the value of H 2 is 

H 2 = — + m — + /-j ...... (61)' 

From this, subtracting the preceding equation, there is found 

*- H >=^% / 62 > 

The second member of this formula is the head lost in friction 
in the length / 2 — l x (Art. 74), and the first member is the 
difference of the piezometer elevations. Thus is again proved 
the principle of Art. 70, that the difference of two piezometer 
elevations shows the head lost in the pipe between them ; in 
Art. 70 the elevations H x and H 2 were measured upward from 
the datum plane, while here they are measured downward. 

By the help of this principle the velocity of flow in a pipe 
may be approximately determined. A line of levels is run 
between the points D x and D 2 , which are selected so that no 
sharp curves occur between them, and thus the difference 
H 2 — H x is found ; / 2 — l x , or the length between D x and D, y 



1 88 FLOW THROUGH PIPES. [Chap. VII. 

is ascertained by careful chaining. Then, from the above 
formula, 



from which v can be computed by the help of the friction fac- 
tors in the table of Art. 74. For example, STEARNS, in 1880, 
made experiments on a conduit pipe 4 feet in diameter under 
different velocities of flow.* In experiment No. 2 the length 
K — h was I 747- 2 feet, and the difference of the piezometer 
levels was 1.243 f eet - Assuming for /the mean value 0.02, and 
using 32.16 for^*, the velocity was 

. / 64.32 X I.243 X 4 , , 

v — \ — — — = 3.0 feet per second. 

V 0.02 x 1747 

This velocity in the table of friction factors gives/" = 0.01 5 for 
a 4-foot pipe. Hence, repeating the computation, there is 
found v = 3.50 feet per second ; it is accordingly uncertain 
whether the value of /is 0.015 or 0.014. If the latter value be 
used there is found 

v = 3.62 feet per second. 

The actual velocity, as determined by measurement of the 
water over a weir, was 3.738 feet per second, which shows that 
the computation is in error about 4 per cent. 

The gauging of the flow of a pipe by piezometers is liable 
to give defective results, partly because the piezometer may 
not indicate the mean pressure in the pipe owing to an imper- 
fect manner of connection, and partly because the formula for 
computing the velocity is merely an empirical one. The dif- 
ference 77 2 — H 1 in order to be reliable should be taken at 

* Transactions of American Society of Civil Engineers. 1SS5. vol. xiv. p. 1. 



Art. 84.] THE HYDRAULIC GRADIENT. 1 89 

points as far apart as possible, and care be taken that no losses 
of head occur between them except that due to friction. Easy 
curves give no perceptible loss of head and need not be con- 
sidered, but obstructions in the pipe or changes in section may 
render the measurement valueless. When pressure gauges 
are used, as must be often the case under high heads, care 
should be taken to test them before making the experiment. 
The pressure gauges, as generally graduated, give the pres- 
sures in pounds per square inch. If then the readings ^ and 
p 2 are taken at D x and D 2 , the pressure-heads in feet are 

*, = 2.304/, and h 2 = 2.304/,. 

The vertical distances A 1 D 1 and AJD 2 having been previously 
determined by levels, the heads H 1 and H 2 are 

~H X = AiD x - h\ and H, = A 2 D 2 - h„ 

from which H 2 — H x is known. Or if the vertical fall z be- 
tween D x and Z> 2 is determined, 

H\ — H x = k x -k 2 +z, 

which is the loss of head between D x and D 2 

Prob. 106. At a point 500 feet from the reservoir, and 28 
feet below its surface, a pressure gauge reads 10.5 pounds per 
square inch ; at a point 8500 feet from the reservoir and 280.5 
feet below its surface, it reads 61 pounds per square inch. 
Show that the discharge per second is about 6 cubic feet if the 
pipe be 12 inches in diameter. 

Article 84. The Hydraulic Gradient. 

The hydraulic gradient is a line which connects the water 
levels in piezometers placed at intervals along the pipe ; or 
rather, it is the line to which the water levels would rise if 
piezometer tubes were inserted. In Fig. 56 the line BC is the 



190 



FLOW THROUGH PIPES. 



[Chap. VI i. 



hydraulic gradient, and it is now to.be shown that for a pipe 
of uniform size this is approximately a straight line. For a 
pipe discharging freely into the air, as in Fig. 56, this line joins 
the outlet end with a point B near the top of the reservoir. 
For a pipe with submerged discharge, as in Fig. 57, it joins the 
lower water level with the point B. 

Let D 1 be any point on the pipe distant l x from the reser- 
voir, measured along the 
pipe line. The piezometer 
there placed rises to C 1 , 
which is a point in the 
hydraulic gradient. The 
equation of this line with 
Fig. 57 ■ reference to the origin A is 

given by the formula of the preceding article, 



H x = (1 + m) - 




d 2jr' 



in which H l is the ordinate A x C xi and l x is the abscissa AA lt pro- 
vided that the length of the pipe is sensibly equivalent to its 

horizontal projection. In this equation the term (1 -|- m) — is 

constant for a given velocity, and is represented in the figure 
by AB or A X B X ; the second term varies with l x , and is repre- 
sented by B X C X . The gradient is therefore a straight line, sub- 
ject to the provision that the pipe is laid approximately hori-- 
zontal ; which is usually the case in practice, since quite mate- 
rial vertical variations may exist in long pipes without sensibly 
affecting the horizontal distances. 

When the variable point D x is taken at the outlet end of 
the pipe, H l becomes the head h, and /, becomes the total 
length /, agreeing with the formula of Art. 76. if the losses of 
head due to curvature and valves be omitted. When D x is 



Art. 84.] 



THE HYDRAULIC GRADIENT. 



I 9 I 



taken very near the inlet end, / becomes zero and the ordinate 
H x becomes AB, which represents the velocity-head plus th& 
loss of head at entrance. 



When easy horizontal curves exist, the above conclusions 
are unaffected, except that the gradient BC is always vertically 
above the pipe, and therefore can be called straight only by 
courtesy, although as before the ordinate B l C 1 is proportion^ 
to /, . When sharp curves exist, the hydraulic gradient is de- 
pressed at each curve by an amount equal to the loss of head 
which there occurs. 




If the pipe -is so laid that a portion of it rises above the hy- 
draulic gradient as at D x in Fig. 58, an entire change of condi- 
tion generally results. If 
the pipe be closed at C, all 
the piezometers stand in 
the line A A, at the same 
level as the surface of the 
reservoir. When the valve 
at C is opened, the flow at Fig. 58. 

first occurs under normal conditions, h being the head and BC 
the hydraulic gradient. The pressure-head at D x is then neg- 
ative, and represented by D 1 C r This results in a partial vacu- 
um in that portion of the pipe whereby the continuity of the 
flow is broken, and as a consequence the pipe from D x to C is 
only partly filled with water. The hydraulic gradient is then 
shifted to B.D 1 , the discharge occurs at D 1 under the head 
A l D 1 , while the remainder of the pipe acts merely as a channel 
to deliver the flow. It usually happens that this change re- 
sults in a great diminution of the discharge, so that it has 
often been necessary to dig up and relay portions of a pipeline 
which have been inadvertently run above the hydraulic gra- 
dient. This trouble can always be avoided by preparing a 



192 FLOW THROUGH PIPES. [Chap. VI L 

profile of the proposed route, and drawing the hydraulic gra- 
dient upon it. 

When a large part of a pipe lies above the hydraulic gradi- 
ent it is called a siphon. Conditions sometimes exist which 
require the construction of siphons, and to insure their suc- 
cessful action pumps must be attached near the highest eleva- 
tions, which may be occasionally operated to remove the air 
that has accumulated, and which would otherwise cause the 
flow to diminish and ultimately to cease. 

The pressure-head, or piezometer height h x , at any point of 
the pipe can be computed if the velocity of flow is known, as 
also the depth H of that point below the water surface in the 
reservoir. In the above figures the ordinate A 1 D 1 is the depth 
H. Then 



K=H-(i +„+/§* 



*g 

in which v must be known by measurement or be computed by 
the method of Art. 76 from the total length / and the given 
head h. This may be put into a simpler form by substituting 
for v its value in terms of / and /i, which gives 

1 + « +A 

h x = H-- jh; (63) 

or for long pipes, where 1 -f- m may be neglected, 

h=H- l fi.. : (6 3 y 

This formula, indeed, can be directly derived from the above 
figures by similar triangles, taking the point B as coincident 
with A y which for long pipes is allowable, since AB is very 
small (Art. 80). 



Art. 85.] A PIPE WITH A NOZZLE. I93 

The above discussion shows that it is immaterial where the 
pipe enters the reservoir, provided that it enters below the 
hydraulic gradient point B. It is also not to be forgotten that 
the whole investigation rests on the assumption that the lengths 
/, and / are sensibly equal to their horizontal projections. 

Prob. 107. A pipe 3 inches in diameter discharges 538 cubic 
feet per hour under a head of 12 feet. At a distance of 300 
feet from the reservoir the depth of the pipe below the water 
surface in the reservoir is 4.5 feet. Compute the probable 
pressure-head at this point. Ans. — 0.2 feet. 

Article 85. A Pipe with a Nozzle. 

Water is often delivered through a nozzle in order to per- 
form work upon a motor or for the purposes of hydraulic min- 
ing, the nozzle being; attached 

s ' s _Ai__ __A 1 

to the end of a pipe which f 1 

brings the flow from a reser- j 

voir. In such a case it is de- 7 : 

sirafMe that the pressure at the ! / 

entrance to the nozzle should t ^ 'O - — -^r 

■<- — - — 

be as great as possible, and Fig. 59. 

this will be effected when the loss of head in the pipe is as 
small as possible. The pressure column in a piezometer, sup- 
posed to be inserted at the end of the pipe, as shown at CJD X 
in Fig. 59, measures the pressure-head there acting, and the 
height A 1 C 1 measures the lost head plus the velocity-head, the 
latter being very small. 

Let h be the total head on the end of the nozzle, l x its 
length, d x its diameter, and v 1 the velocity of discharge at the 
small end. Let /, d, and v be the corresponding quantities for 
the pipe. Then the effective velocity-head of the issuing stream 

is — , and the lost head is h . This lost head consists of 

*g 2 g 



194 FLOW THROUGH PIPES. [Chap. VII. 

several parts — that lost at the entrance D 1 ; that lost in friction 
in the pipe ; that lost in curves and valves, if any ; and lastly, that 
lost in the nozzle. Thus, 

v: ■ i? i * v\ v? 

2g 2g y d 2g 2g >2g 

Here m is determined by Art. 73, /by Art. 74, n by Art. 75, 
and m x is to be found from the coefficient of velocity of the 
nozzle (Art. 63) in the same manner as m. If, for instance, c. 
for the nozzle is 0.98, then 

and for a perfect nozzle vi x would be zero. „The value of m y 
includes all losses of head in the nozzle, as m does in the en- 
trance tube, so that the length l x need not be considered. 

The velocities v and v 1 are inversely as the areas of the cor- 
responding sections, whence 

d 2 

Inserting this in the above expression, and solving for v, gives 
the formula 



/ ^ Orjl' 



V 



d ' v ' 'V; 

from which v can be computed by the tentative method ex- 
plained in Art. /6. This equation, in connection with the pre- 
ceding, shows that the greatest velocity v 1 obtains when d is as 
large as possible compared to d 1 . As the object of a nozzle is 
to utilize either the velocity or the energy of the water, a large 
pipe and a small nozzle should hence be employed to gtve the 
best result, and this is attained when the velocity z\ is nearly 
equal to V2gh, 



Art. 85.] A PIPE WITH A NOZZLE. 1 95 

As a numerical example, the effect of attaching a nozzle to 
the pipe whose discharge was computed in Art. jy will be con- 
sidered. There /= 500, ^ = 0.25, and /* = 4 feet; ^=0.5, 
n = o, ^ = 2.15 feet; and ^ = 0.106 cubic feet, per second. 
Now let the nozzle be one inch in diameter at the small end, or 
d 1 — 0.0833 f eet and c l = 0.98, whence m x = 0.041. Using 
f= 0.029, the velocity in the pipe is 



2 X 32.16 X 4 

v = 



0.5+0.029 X 500 X 4+ 1-041 X 81 

whence 27=1.35 feet per second. The effect of the nozzle, 
therefore, is to reduce the velocity, owing to the loss of head 
which it causes. The velocity of flow from the nozzle is 

v x = 1.35 X 9 = 12.15 f eet P er second ; 

and the discharge per second is 

q — 0.7854 X O.25 2 X 1-35 = O.066 cubic feet . 

which is about 40 per cent less than that of the pipe before the 
nozzle was attached. The nozzle, however, produces a marvel- 
lous effect in increasing the energy of the discharge ; for the 
velocity-head corresponding to 2.15 feet per second is only 
0.072 feet, while that corresponding to 12.15 feet per second is 
2.30 feet, or about 32 times as great. As the total head is 4 
feet, the efficiency of the stream issuing from the nozzle is 
about 57 per cent. 

If the pressure-head h x at the entrance of the nozzle be 
observed, either by a piezometer or by a pressure gauge, the 
velocity of discharge can be computed by the formula 



v. 



ay 



whose demonstration is given in Art. 63. 



196 



FLOW THRO UGH PIPES. 



[Chap. VII. 



Prob. 108. A pipe 3 inches in diameter and 800 feet long" 
runs from a reservoir to a nozzle which drives a water motor. 
There are no curves or valves in the pipe and no loss of head 
at entrance or in the nozzle. The head on the nozzle is 100 
feet. Find the diameter of the nozzle which will furnish the 
maximum discharge. Also the diameter which will give the 
maximum horse-power. (See Art. 150.) 




Article 86. House Service Pipes. 

A service pipe which runs from a street main to a house is 
connected to the former at right angles, and usually by a 
" ferrule" which is smaller in diameter than the pipe itself. 

The loss of head at entrance 
is hence larger than in the 
cases before discussed, and m 
should probably be taken as 
at least equal to unity. The 
pipe, if of lead, is frequently 
carried around sharp corners 
Fig. 60. by curves of small radius; if 

of iron, these curves are formed by pieces forming a quadrant 
of a circle into which the straight parts are screwed, the radius 
of the centre line of the curve being but little larger than the 
radius of the pipe, so that each curve causes a loss of head 
equal nearly to double the velocity-head (Art. 75). For new 
clean pipes the loss of head due to friction may be estimated 
by the rules of Art. 74. 

A water main should be so designed that a certain minimum 
pressure-head h x exists in it at times of heaviest draught. This 
pressure-head may be represented by the height of the piezom- 
eter column AB, which would rise in a tube supposed to be 
inserted in the main, as in Fig. 60. The head // which causes 
the flow in the pipe is then the difference in level between 



Art. 86.] HOUSE SERVICE PIPES. 1 97 

the tup of this column and the end of the pipe, ox AC. In- 
serting for h this value, the formulas of Arts. 76 and jj may be 
applied to the investigation of service pipes, in the manner 
there illustrated. As the sizes of common house-service pipes 
are regulated by the practice of the plumbers and by the market 
sizes obtainable, it is not often necessary to make computations 
regarding them. 

The velocity of flow in the main has no direct influence upon 
that in the pipe, since the connection is made at right angles. 
But as that velocity varies, owing to the varying draught upon 
the main, the pressure-head k t is subject to continual fluctua- 
tions. When there is no flow in the main, the piezometer 
column rises until its top is on the same level as the surface of 
the reservoir ; in times of great draught it may sink below C, so 
that no water can be drawn from the service pipe. 

The detection and prevention of the waste of water by con- 
sumers is a matter of importance in cities where the supply 
is limited and where meters are not in use. Of the many 
methods devised to detect this waste, one by the use of pie- 
zometers may be noticed, by which an inspector without enter- 
ing a house may ascertain whether water is being drawn within, 
and the approximate amount per second. Let Mbe the street 
main from which a service pipe MOH runs to a house H. At 
the edge of the sidewalk a tube OP is connected to the service 
pipe, which has a three-way cock at O, which 
can be turned from above. The inspector, h 
passing on his rounds in the night-time, at- 
taches a pressure gauge at P and turns the 
cock O so as to shut off the water from the ^ 
house and allow the full pressure of the main FlG - 6l - 

p x to be registered. Then he turns the cock so that the water 
may flow into the house, while it also rises in OP and registers 
the pressure^. Then if / 2 is less than/, it is certain that a 




I98 FLOW THROUGH PIPES. [Chap. VIL 

waste is occurring within the house, and the amount of this may 
be approximately computed if desired, in the manner indicated 
in Art. 70, and the consumer be fined accordingly.* 

Prob. 109. Describe a water-pressure regulator to be placed 
between the main and the house so that the pressure in the 
service pipes may never exceed a given quantity — say 40 pounds 
per square inch. 

Article 87. A Water Main. 

The simplest case of the distribution of water is that where 
a single main is tapped by a number of service pipes near its 
£nd, as shown in Fig. 62. In designing such a main the prin- 
cipal consideration is that it should 
==s be large enough so that the pres- 
EE sure-head h x , when all the pipes 

— are in draught, shall be amply suf- 

— ficient to deliver the water into 





J 

J l 






■f I 

I' 1 


f 


I 


0*0 O CP 


' 



<, ^._ ^ — - ? - — 3 the highest houses along the line. 

Fig. 62. Fanning recommends that this 

pressure-head in commercial and manufacturing districts should 
not be less than 150 feet, and in suburban districts not less 
than 100 feet. The height H to the surface of the water in 
the reservoir will always be greater than h x , and the pipe is to 
be so designed that the losses of head may not reduce A, 
below the limit assigned. The head h to be used in the for- 
mulas is the difference H — h x . The discharge per second q 
being known or assumed, the problem is to determine the 
diameter d of the main. 

A strict theoretical solution of even this simple case leads 

to very complicated calculations, and in fact cannot be made 

without knowing all the circumstances regarding each of the 

service pipes. Considering that the result of the computation 

* This briefly describes Church's water-waste indicator. 



Art. 87.] A WATER MAIN. 1 99 

is merely to enable one of the market sizes to be selected, it is 
plain that great precision cannot be expected, and that ap- 
proximate methods may be used to give a solution entirely 
satisfactory. It will then be assumed that the service pipes 
are connected with the main at equal intervals, and that the 
discharge through each is the same under maximum draught. 
The velocity v in the main then decreases, and becomes o at 
the dead end. The loss of head per linear foot in the length 
l x (Fig. 62) is hence less than in /. To estimate this, let v 1 be 
the velocity at a distance x from the dead end ; then 

x 

v x = j v. 

The loss of head in friction in the length dx is 

$*< x 1 v 2 n 

Sh" =f-r — = f-jn — $x; 
J d 2g J dl; 2g 

and hence between the limits o and l x that loss is 

h "=fhi' (65) 

provided that/" remains constant. This is really not the case, 
but no material error is thus introduced, since /must be taken 
larger than the tabular values in order to allow for the deteri- 
oration of the inner surface of the main. The loss of head in 
friction for a pipe which discharges uniformly along its length 
may therefore be taken at one-third of that which occurs when 
the discharge is entirely at the end. 

Now neglecting the loss of head at entrance and the effec- 
tive velocity-head of the discharge, the total head h is entirely 
consumed in friction, or 

y d 2g J 3d 2g 



200 FLOW THROUGH PIPES. [Chap. VII. 

Placing in this for v its value in terms of the total discharge q, 
and solving for d, gives 



v l d iy 2gn 2 h 

This is the same as the formula of Art. 80, except that / has 
been replaced by / -4- J-/ a . The diameter in feet then is 

d = 0.479 ( / + K) i (x)*> 

as in the case of long pipes. 

For example, consider a village consisting of a single street, 
whose length /, = 3000 feet, and upon which there are 100 
houses, each furnished with a service pipe. The probable 
population is then 500, and taking 100 gallons per day as the 
consumption per capita, this gives the average discharge per 

second 

500 X 100 t . f 

q = -= = O.0774 cubic feet ; 

1 7.48 X 3600 X 24 ' '^ 

and as the maximum draught is often double of the average, 
q will be taken as 0.15 cubic feet per second. The length / 
to the reservoir is 4290 feet, whose surface is 90.5 feet above 
the dead end of the main, and it is required that under full 
draught the pressure-head in the main shall be 75 feet. Then 
h — 90.5 — 75 = 15.5 feet, and taking/" = 0.03 in order to be 
on the safe side, the formula gives 

d = 0.36 feet = 4.3 inches. 

Accordingly a four-inch pipe is nearly large enough to satisfy 
the imposed conditions. 

To consider the effect of fire service upon the diameter of 
the main, let there be four hydrants placed at equal intervals 
along the line l x , each of which is required to deliver 20 cubic 
feet per minute under the same pressure-head of 75 feet. This 
gives a discharge 1.33 cubic feet per second, or, in total, 



Art. 



!•] 



A MAIN WITH BRANCHES. 



20 1 



q = 1.33 -j- 0.15 = 1.5 cubic feet. Inserting this in the for- 
mula, and using for/" the same value as before, 
d — 0.897 feet == 10.8 inches. 

Hence a ten-inch pipe is at least required to maintain the 
required pressure when the four hydrants are in full draught at 
the same time with the service pipes. 

Prob. no. Compute the velocity v. and the pressure-head h^ 
for the above example, if the main be 10 inches in diameter 
and the discharge 1.5 cubic feet per second. 



Article 88. A Main with Branches. 

In Fig. 63 is shown a main of length / and diameter d, 
having two branches with lengths / 2 and / 2 , and diameters d z 




Fig. 63. 

and d 2 . These being given, as also the heads H x and H 2 under 
which the flow occurs, it is required to find the discharges q x and 
q^ . Let v, v x , and v 2 be the corresponding velocities ; then for 
long pipes, in which all losses except those due to friction may 
be neglected, 



H x -y=L 



d. 



2g 



H t -y 



f k*± 



where y is the difference in level between the reservoir surface 
and the water level in a piezorneter supposed to be inserted at 
the junction. This y is the friction-head consumed in the large 
main, or 

y=f dT g - 



202 FLOW THROUGH PIPES. [Chap. VII. 

Inserting this in the two equations, and placing for the veloci- 
ties their values in terms of the discharges, they become 






/, 


^7 


?/ 


/, 


i. 


q- 



(66) 



16 2 J d l 

*rom which q 1 and q 2 are best obtained by trial ; although by 
solution the value of each may be directly expressed in terms 
of the given data, the expressions are too complicated for 
general use. 

When it is required to determine the diameters from the 
given lengths, heads, and discharges, there are three unknown 
quantities, d, d 1 , d^ , to be found from only two equations, and 
the problem is indeterminate. If, however, d be assumed, 
values of d 1 and d 2 may be found ; and as d may be taken at 
pleasure, it appears that an infinite number of solutions is pos- 
sible. Another way is to assume a value of y, corresponding 
to a proper pressure-head at the junction ; then the diameters 
are directly found from the usual formula for long pipes, 

d = 0.479 [-j-), 

in which h is replaced by y for the large main, and by U 1 — y 
and H 2 — y for the smaller ones, q for the first being q 1 -\-q i ,' 
and for the others q 1 and q 2 respectively. 

A water-supply system consists of a principal main with many 
sub-mains as branches. In designing these the quantities of 
water to be furnished are assumed from the present and prob- 
able future population, which in small towns requires from 40 
to 100 gallons per capita per day. and in large cities from 100 
to 150 gallons. This should be furnished under heads sufficient 
to raise the water into the highest houses, as also for use in 



Art. 89.] PUMPING THROUGH PIPES. 203 

cases of fire. As the problem of computing the diameters 
from the given data is indeterminate, it will probably be as well 
to assume at the outset the sizes for the principal mains. The 
velocities corresponding to the given quantities and the as- 
sumed sizes are then computed, and from these the pressure- 
heads at a number of points are found. If these are not satis- 
factory, other sizes are to be taken and the computation be 
repeated. The successful design will be that which will furnish 
the required quantities under proper pressures with the least 
expenditure. 

Prob. in. In Fig. 63 let q x = 0.5 and q 2 = 0.4 cubic feet; 
^=140 and i/ 2 =i25 feet; / : = 3810, 4=2455, and / = 
12 314 feet. If d x equals^ find the values of d and d x , and 
also the pressure-head at the junction if its depth below the 
reservoir level is 108 feet. 



Article 89. Pumping through Pipes. 

When water is pumped through a pipe from a lower to a 
higher level, the power of the pump must be sufficient not only 
to raise the required amount in a given time, but also to over- 
come the various resistances to flow. The head due to the re- 
sistances is thus a direct source of loss, and it is desirable that 
the pipe be so arranged as to render this as small as possible. 

Let w be the weight of a cubic foot of water and q the 
quantity raised per second through the height H, which, for 
example, may be the difference in level 
between a canal C and a reservoir R, as 
in Fig. 64. The useful work done by 
the pump in each second is wqH. Let h! 
be the head lost in entering the pipe at 
the canal, h" that lost in friction in the 
pipe, and h!" all other losses of head, such as those caused by 




204 FLOW THROUGH PIPES. [Chap. VII. 

curves, valves, and by resistances in passing through the pump 
cylinders. Then the total work performed by the pump per 
second is 

& = wgH-+wg(tf + /k"'+.M"). . . . (67) 

Inserting; for the lost heads their values, this becomes 



/ 



k = tvgff + fvg[m+f2+n)-. • • • (67)' 

In order, therefore, that the losses of work may be as small as 
possible, the velocity of flow through the pipe should be low ; 
and this is to be effected by making the diameter of the pipe 
large. 

For example, let it be required to determine the horse- 
power of a pump to raise 1 200 000 gallons per day through a 
height of 230 feet, when the diameter of the pipe is 6 inches 
and its length 1400 feet. The discharge per second is 

I 200 000 c . , . £ , 

= 1.86 cubic feet, 



7.481 X 24 X 3 6 oo 
and the velocity of flow is 

v = — = r, = Q.47 feet per second. 

0.7854 X 0.5 2 y ^ 7 F 

The probable head lost at entrance into the pipe is 
ti — 0.5— = 0.5 X 1-39 = 0.7 feet. 

2g 

When the pipe is new and clean the friction factor f is 
about c.020, as shown by Table XVI ; then the loss of head 
in friction is 

k" = 0.020 X — ~ X 1-39 = 77-3 feet. 
0.5 



Art. 3 9 .] PUMPING THROUGH PIPES. 205 

The other losses of head depend upon the details of the valve 
and pump cylinder; if these be such that n = 4, then 

h!" — 4 X 1.39 = 5-6 feet. 

The total losses of head hence are 

h> J^k" + h!" = 84.1 feet. 

The work to be performed per second by the pump now is 

k — 62.5 X 1.86(230 + 84.1) = 36 510 foot-pounds, 

and the horse-power expended is 

36510 



HP = ^^-- = 66.4. 

If there were no losses in friction and other resistances the 
work done would be simply 

k = 62.5 X 1.86 X 230 = 26 740 foot-pounds, 

and the corresponding horse-power would be 

s?= 26740 6> 

55o 

Accordingly 17.8 horse-power is wasted in injurious resistances. 

For the same data let the 6-inch pipe be replaced by one 
14 inches in diameter. Then, proceeding as before, the velocity 
of flow is found to be 1.80 feet per second, the head lost at 
entrance 0.03 feet, the head lost in friction 1.23 feet, and that 
lost in other ways 0.20 feet. The total losses of head are thus 
only 1.46 feet, as against 84.1 feet for the smaller pipe, and the 
horse-power required is 48.9, which is but little greater than 
the theoretic power. The great advantage of the larger pipe 
is thus apparent, and by increasing its size to 18 inches the 
losses of head may be reduced so low as to be scarcely appre- 
ciable in comparison with the useful head of 230 feet. 

A pump is often used to force water directly through the 



206 



FLOW THROUGH PIPES. 



[Chap. VII. 




mains of a water-supply system under a designated pressure. 

The work of the pump in this 

case consists of that required to 

maintain the pressure and that 

required to overcome the fric- 

tional resistances. Let h x be the 

pressure-head to be maintained 

at the end of the main, and z the 

height of the main above the 

FlG - 6 5- level of the river from which the 

water is pumped ; then h x -f- z is the head H, which corresponds 

to the useful work of the pump, and, as before, 

k = wqH + wq{h! + h" + h'"). 

To reduce these injurious heads to the smallest limits the 
mains should be large in order that the velocity of flow may 
be small. In Fig. 65 is shown a symbolic representation of 
the case of pumping into a main, P being the pump, C the 
source of supply, and DM the pressure-head which is main- 
tained upon the end of the pipe during the flow. At the 
pump the pressure-head is AP, so that AD represents the hy 
draulic gradient for the pipe from P to M. The total work of 
the pump may then be regarded as expended in lifting the water 
from C to A, and this consists of three parts, corresponding to 
the heads CM or z, MD or h x , and AB or h! + h" + h"\ the 
first overcoming the force of gravity, the second delivering 
the flow under the required pressure, while the last is trans- 
formed into heat in overcoming friction and other resistances. 
In this direct method of water supply a standpipe.^-i/ 5 , is often 
erected near the pump, in which the water rises to a height 
corresponding to the required pressure, and which furnishes 
a supply when a temporary stoppage of the pumping engine 
occurs. 

Prob. 112. Compute the horse-power of a pump for the fol- 



Art. qo.] FIRE HOSE. 2 ^7 

lowing data, neglecting all resistances except those due to fric- 
tion : q = 1.5 cubic feet per second, which is distributed uni- 
formly over a length /, = 3000 feet, the remaining length of 
the pipe being 4290 feet ; d = 10 inches, H = 86.4 feet. 

Article 90. Fire Hose. 

Fire hose is generally 2\ inches in diameter, and lined with 
rubber to reduce the frictional losses. The following values 
of the friction factor f have been deduced from the experi- 
ments of Freeman :* 

Velocity in feet per second, v>— 4 6 10 15 20 

Unlined linen hose, f — 0.038 0.038 0.037 0.035 °-°34 

Rough rubber-lined cotton, f= 0.030 0.031 0.031 0.030 0.029 

Smooth rubber-lined cotton, f— 0.024 0.023 0.022 0.019 0.018 

Discharge, gallons per minute = 61 92 153 230 306 

By the help of this table computations may be made on the 
pumping of water through hose for delivery in fire streams or 
for other purposes, in the same manner as for pipes. It is seen 
that the friction factors for the best hose are slightly less than 
those given for 2j-inch pipes in Table XVI. 

The loss of head in a long hose becomes so great even 
tinder moderate velocities as to consume a large proportion of 
the pressure exerted by the hydrant or steamer. For example, 
let this primitive pressure be 122 pounds per square inch, cor- 
responding to a head of 281 feet, and let it be required to find 
the pressure-head in 2-J-inch rough rubber-lined cotton hose at 
1000 feet distance, when a nozzle is used which discharges 153 
gallons per minute, the hose being laid horizontal. In cubic 
feet per second the discharge is 

153 



7.48 X 60 



0.341 = 



* Hydraulics of Fire Streams. Transactions American Society of Civil 
Engineers, November, 1889. 



208 FLOW THROUGH PIPES. [Chap. VII, 

and the velocity in the hose is accordingly found to be 

v = t ^ = 10.0 feel; per second 
%nd" 

Hence the loss of head in friction is 

/ v* 
h" —f-j — = 231 feet, 

and consequently the pressure-head at the entrance to the 
nozzle is 

A 1 = 281 — 231 = 50 feet, 

which corresponds to about 22 pounds per square inch. The 
remedy for this great reduction of pressure is to employ a 
smaller nozzle, thus decreasing the discharge and the velocity 
in the hose; but if both head and quantity of discharge are 
desired they can only be secured either by an increase of pres- 
sure at the steamer or by the use of a larger hose. 

Prob. 113. When the pressure gauge at the steamer indi- 
cates 83 pounds per square inch, a gauge on the leather hose 
800 feet distant reads 25 pounds. Compute the value of the 
friction factor/", the discharge per minute being 121 gallons. 

Ans. 0.036. 

Art. 91. Lampe's Formula. 

There have been made many attempts to express the mean 
velocity v without the use of a factor or coefficient of friction. 
That this can be empirically done, within the range of experi- 
mental results, is plain by observing that the values of f in 
Table XVI show a regular variation with the diameter d. For 
long pipes/" is then a function of d and z\ or a function of d, //. 
and /. The simplest expression of the relation between these 
quantities is 



Art. 9 1.] LAM PES FORMULA. 209 

in which a, f3, and y are empirical constants. The investiga- 
tions of Lampe have determined probable values for these 
constants, giving 

-^0.555 

7> 



77'7d^\-j) , (68) 



in which d, //, and / are to be taken in feet, and v will be 
found in feet per second, This formula is only applicable to 
long circular pipes with surfaces clean or in fair condition. 

From this formula the discharge q may be expressed 

//A - 555 
q = 6i.od 2 ^[j) , (68)' 

and the diameter required to discharge a given quantity is 

//N 0.206 

d = 0.217 t zl \-i] ...... (68) /; 

By the use of these formulas all of the preceding problems 
concerning long pipes may be directly solved without the use 
of the tables of friction factors. They show that the discharg- 
ing capacity of long pipes varies about as the 2.7 power of the 
diameter (Art. 80). 

As an example, let it be required to find the diameter of a 
pipe which is to discharge 177 300 gallons per hour, its length 
being 75 000 feet and the head 135 feet. Here 

177 300 „ , . - 

q = — ^— ~ -- = 6.583 cubic feet, 

* 3600 X 7.481 D J ' 

/ 75 000 

and t — ~ =555-6; 

h 135 ^ 

whence by the formula 

d= o.2i7(6.583) a37I (555.6) - 206 , 
which gives 

d= 1. 61 feet == 19.3 inches, 

so that a 20-inch pipe should be selected. 

Prob. 114. Solve Problems 102 and 103 by the use of the 
above formulas. 



2I ° FLOW THROUGH PIPES. [Chap. VII. 

Article 92. Very Small Pipes. 

The preceding investigations and rules apply to pipes greater 
than about 0.5 inches in diameter, and are not valid for very 
small pipes. The laws of discharge in these are not understood 
from a theoretical basis, but experiments made by POISEUILLE 
in 1843, m order to study the phenomena of the flow of blood 
in veins and arteries, have settled beyond question that they 
are materially different from those which govern large pipes at 
ordinary velocities. His investigations proved that for pipes 
whose diameters are less than about 0.7 millimeters or 0.03 
inches, the velocity is expressed by the simple relation 

hd 2 
v = a-, 

in which a is a factor nearly constant at a given temperature. 
The velocity then varies directly with the head and with the 
square of the diameter, and inversely with the length. It is 
here supposed that the pipe is long, so that losses of head due 
to entrance may be neglected. 

Later researches indicate that these laws are also true for 
large pipes, provided the velocity be small ; and that for a 
given pipe there is a certain critical velocity at which the law 
changes, and beyond which 



\/ hd 



as for the case of common pipes. This critical point appears 
to be that at which the filaments cease to move in parallel 
lines, and pass in sinuous paths from one side of the pipe to 
the other. For a very small pipe the velocity may be high 
before this point is reached ; for a large pipe it happens at very 
low velocities. 



Art. 92.] VERY SMALL PLPES. 21 X 

In Art. 74 it was mentioned that the frictional resistances 
in a pipe consist of those along the inner surface, and of those 
met among the particles in their sinuous motion. Since in 
small pipes the latter do not exist, it appears from PoiSEUILLE's 
formula that the head lost in friction along the inner surface 
may be expressed by 

ad 2 

Now if the law were known which governs the loss in internal 
friction it might be possible to add this to the preceding, and 
thus obtain an expression for loss of head in which the friction 
factor would be a quantity dependent only upon the nature of 
the surface. Thus far, however, efforts in this direction have 
not been practically successful. 

The effect of temperature on the flow has not been consid- 
ered in the previous articles, and in fact but little is known re- 
garding ir, except that a very slight increase in discharge is prob- 
able for a high rise in the temperature of the water. For very 
small pipes, however, POISEUILLE found that a marked in- 
crease in velocity and discharge resulted, the value of a being 
about twice as great at 45 Centigrade as at io°. 

Prob. 115. The value of a for small pipes is about 184 
when h, d, /, and v are in millimeters, and the flow occurs at 
io° Centigrade. Find its value when the foot is the unit of 
measure. 



12 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 



CHAPTER VIII. 
FLOW IN CONDUITS AND CANALS. 

Article 93. Definitions. 

Water is often conveyed from place to place in artificial 
channels, such as troughs, aqueducts, ditches, and canals, there 
being no head to cause the flow except that due to the slope. 
The word conduit will be used as a general term for a channel 
lined with timber, mortar, or masonry, and will also include 
metal pipes, troughs, and sewers. Conduits may be either 
open as in the case of troughs, or closed as in sewers and most 
aqueducts. Streams flow in natural channels eroded in the 
earth, and include small brooks as well as the largest rivers. 
Most of the principles relating to conduits and canals apply 
also to streams, and the word channel will be used as applica- 
ble to all classes. 

The wetted perimeter of the cross-section of a channel is 
that part of its boundary which is in contact with the water. 
Thus, if a circular sewer of diameter d be half full of water 
the wetted perimeter is \~d. In this chapter the letter/; will 
designate the wetted perimeter. 

The hydraulic radius of a water cross-section is its area 

divided by its wetted perimeter. Let a be the area and r the 

hydraulic radius ; then 

a 
r = — . 

P 

The letter r is of frequent occurrence in formulas for the flow 




Art. 93.] DEFINITIONS. 213 

in channels ; it is a linear quantity, which is always expressed 
in the same unit as p. It 
is also frequently called 
the hydraulic depth or hy- 
draulic mean depth, be- FlG ' 66 ' 
cause for a shallow section its value is but little less than the 
mean depth of the water. Thus in Fig. 66, if b be the breadth 
on the water surface, the mean depth is a -f- b, and the hy- 
draulic radius is a -=- p; and these are nearly equal, since p is 
but slightly larger than b. 

The hydraulic radius of a circular cross-section filled with 
water is one-fourth of the diameter; thus: 

The same value is also applicable to a circular section half 
filled with water, since then both area and wetted perimeter 
are one-half their former values. 

The slope of the water surface in the longitudinal section, 
designated by the letter s, is the ratio of the fall h to the length 
/ in which that fall occurs, or 

h 
S = J' 

The slope is hence expressed as an abstract number, which is 
independent of the system of measures employed. To deter- 
mine its value with precision h must be obtained by referring 
the water level at each end of the line to a bench mark by the 
help of a hook gauge or other accurate means, the benches be- 
ing connected by level lines run with care. The distance / is 
measured along the inclined channel, and it should be of con- 
siderable length in order that the relative error in h may not 
be large. t 



214 FLOW IN CONDUITS AND CANALS. [Chap. VIIL 

If there be no slope, or s = o, there can be no flow. But if 
there be even the smallest slope the force of gravity furnishes 
a component acting down the inclined surface, and motion en- 
sues. The velocity of flow evidently increases with the slope. 

The flow in a channel is said to be permanent when the 
same quantity of water per second passes through each cross- 
section. If an empty channel be filled by admitting water at 
its upper end the flow is at first non-permanent or variable, for 
more water passes through one of the upper sections per second 
than is delivered at the lower end. But after sufficient time 
has elapsed the flow becomes permanent ; when this occurs the 
mean velocities in different sections are inversely as their 
areas (Art. 19). 

Uniform flow is that particular case of permanent flow 
where all the water cross-sections are equal, and the slope of the 
water surface is parallel to that of the bed of the channel. If 
the sections vary the flow is said to be non-uniform, or variable, 
although the condition of permanency is still fulfilled. In this 
chapter only the case of uniform flow will be discussed. 

The velocities of different filaments in a channel are not 
equal, as those near the wetted perimeter move slower than 
the central ones owing to the retarding influence of friction. 
The mean of all the velocities of all the filaments in a cross- 
section is called the mean velocity v. Thus if v\ ?>", etc., be 
velocities of different filaments, 

v' +v" + etc. f 

v = n ' (6 9> 

in which n is the number of filaments. Let a be the area of 

the cross-section and a that of one of the elementary filaments; 

a 
then n = — ■. , and hence 
a 

av = a'{v' + v" + etc.). 



Art. Q4-] FORMULA FOR MEAN VELOCITY. 215 

But the second member is the discharge q. Therefore the 
mean velocity may be also determined by the relation 

v = ^ (69)' 

a 

The filaments which are here considered are in part imaginary, 
for experiments show that there is a constant sinuous motion 
of particles from one side of the channel to the other. The 
best definition for mean velocity hence is, that it is a velocity 
which multiplied by the area of the cross-section gives the dis- 
charge, or v = q -r- a. 

Prob. 116. Compute the hydraulic radius of a rectangular 
trough whose width is 4.4 feet and depth 2.2 feet. 

Prob. 117. Compute the mean velocity in a circular sewer 
of 4 feet diameter when it is half filled and discharges 120 gal- 
lons per second. 



Article 94. Formula for Mean Velocity. 

When all the wetted cross-sections of a channel are equal, 
and the water is neither rising nor falling, having attained a con- 
dition of permanency, the flow is said to be uniform. This is 
the case in a conduit or canal of constant size and slope whose 
supply does not vary. The same quantity of water per second 
then passes each cross-section, and consequently the mean 
velocity in each section is the same. This uniformity of flow 
is due to the resistances along the interior surface of the chan- 
nel, for were it perfectly smooth the force of gravity would 
cause the velocity to be accelerated. The entire energy of 
the water due to the fall h is hence expended in overcoming 
frictional resistances along the length /. Let W be the weight 
of water per second which passes any cross-section, F the force 
of friction or resistance per square foot of the interior surface 
of the channel, / the wetted perimeter, and v the mean veloci- 



2l6 FLOW IN CONDUITS AXD CANALS [Chap. VIIL 

ty. Now assuming that the friction is uniform over the entire 
inner surface whose area is //, the total resisting force is Fpl, 
and again assuming that the velocity along the surface is the 
same as v, the total resisting work is Fplv. Hence 

Fplv = Wh. 

But the value of W is wav where a is the area of the cross- 
section, and w is the weight of a cubic unit of water ; accord- 
ingly, 

Fpl = wah, 
or 

ah 

pi 

a h 

Here — is the hydraulic radius r, and j is the slope s, and the 

value of F is 

F = zvrs. 

This is an approximate expression for the resisting force of 
friction per square foot of the interior surface of the channel. 

In order to establish a formula for mean velocity the value 
of F must be expressed in terms of v, and this can only be 
done by studying the results of experiments. These indicate 
that F is approximately proportional to the square of the mean 
velocity. Therefore, if c be a constant, 

v = c Vrs (jo) 



This is an empirical expression for the law of variation of the 
mean velocity with the hydraulic radius and slope of the chan- 
nel. The quantity c is a coefficient which varies with the 
degree of roughness of the bed and with other circumstances. 
It is the object of the following articles to state values of c for 
different classes of conduits and canals. 

Another method of establishing the above formula is simi- 



Art. 94.] FORMULA FOR MEAN VELOCITY. 21? 

lar to that used in Art. 74 for pipes. The total head h repre- 
sents the loss of head in friction ; this should vary directly with 
p and /, and it should vary inversely with a, because for a given 
wetted perimeter the friction will be the least for the largest a. 
It should also vary as the square of the velocity. Hence 

a 2g 

in which/"' is an abstract number depending upon the charac- 
ter of the surface. From this the value of v is 

/2s;aJi , — , w 

v= vf¥ = ' (;o) 

in which c is the square root of 2g-±-f. Notwithstanding 
these reasonings the formula cannot be called rational ; it is 
merely an empirical expression whose basis is experiment. 

To determine values of the coefficient c the quantities v, r, 
and s are measured for particular cases, and then c is computed. 
To find r and s linear measurements are alone required. To 
determine v the flow must be gauged either in a measuring 
vessel or by an orifice or weir, or, if the channel be large, by 
floats or other indirect methods described in the next chapter. 
It being a matter of great importance to establish a satisfactory 
formula for mean velocity, thousands of such gaugings have 
been made, and from the records of these the values of the 
coefficients have been deduced. It is found that c lies between 
30 and 160 when v and r are expressed in feet, and that its 
value is subject to variation, not only with the character of the 
surface, but also with the hydraulic radius and slope. 

Prob. 118. Compute the value of c for a circular masonry 
conduit 4 feet in diameter which delivers 29 cubic feet per 
second when running half full, its slope or grade being 1.5 feet 
in 1000 feet. Ans. 119. 



2l8 



FLOW IN CONDUITS AND CANALS. [Chap. VIII. 



Article 95. Circular Conduits, Full or Half Full. 

When a circular conduit of diameter d runs either full or 
half full of water the hydraulic radius is \d t and the formula 
for mean velocity is 

v = c Vrs = c . h Vds. 

The velocity can then be computed when c is known, and for 
this purpose the following table gives Smith's values of c for 

TABLE XVII. COEFFICIENTS FOR CIRCULAR CONDUITS. 



Diameter 






Velocity 


in Feet per Second. 






in 
















Feet. 


1 


2 


3 


4 


6 


10 


15 


I. 


96 


104 


109 


112 


Il6 


121 


124 


i-5 


103 


in 


Il6 


119 


123 


129 


132 


2. 


109 


116 


121 


124 


129 


134 


I 3 S 


2-5 


113 


120 


125 


I2S 


133 


139 


143 


3- 


117 


124 


123 


132 


136 


143 


147 


3-5 


120 


127 


131 


135 


139 


I46 


151 


4- 


123 


130 


134 


137 


142 


I50 


155 


5- 


12S 


134 


139 


142 


147 


155 




6. 


132 


133 


142 


145 


150 






7- 


135 


141 


145 


149 


153 






8. 


137 


143 


I 4 S 


151 









pipes and conduits having quite smooth interior surfaces, and 
no sharp bends.* The discharge per second then is 

q = av = c . \a i ds } 

in which a is either the area of the circular cross-section or one 
half that section, as the case may be. 

To use this table a tentative method must be employed, 



Hydraulics, p. i~\. 



Art. 95.] CIRCULAR CONDUITS, FULL OR HALF FULL. 219 

since c depends upon the velocity of flow. For this purpose 
there may be taken roughly, 

mean c = 125, 
and then v may be computed for the given diameter and slope ; 
a new value of c is then taken from the table and a new v com- 
puted ; and thus, after two or three trials, the probable mean 
velocity of flow is obtained. The value of d must be expressed 
in feet. 

For example, let it be required to find the velocity and dis- 
charge of a semicircular conduit of 6 feet diameter when laid 
on a grade of 0.1 feet in 100 feet. First, 



v — 125 X \ V6 x 0.001 = 4.8 feet. 
For this velocity the table gives 147 for c\ hence 



v — 147 X \ V 0.006 = 5.7 feet. 

Again, from the table c = 150, and 

v — 150 X i V0.006 =5.8 feet. 

This shows that 150 is a little too large; for c= 149.5, v is 
found to be 5.79 feet per second, which is the final result. 
The discharge per second now is 

q — O.7354 X \ X 36 X 5.79 = 81.9 cubic feet, 

which is the probable flow under the given conditions. 

To find the diameter of a circular conduit to discharge a 
given quantity under a given slope, the area a is to be ex- 
pressed in terms of d in the above equation, which is then to 
be solved for d; thus, for a conduit which runs full, 

and for one which is half full 

V/~ i/cJ 



220 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

Here c at first may be taken as 125 ; then d is computed, and 
the approximate velocity of flow is 



v = 



a/854^ 2 ' 

by which a value of c is selected from the table, and the com- 
putation is then repeated until the corresponding values of c 
and v are found to closely agree. 

As an example of the determination of diameter let it be re- 
quired to find d when q = 81.9 cubic feet per second, s = 0.00 1, 
and the conduit runs full. For c = 1 25 the formula gives d = 4.9 
feet, whence v = 4.37 feet per second. From the table c may 
be now taken as 142, and repeating the computation d = 4.64 
feet, whence v = 4.84 feet per second, which requires no further 
change in the value of c. As the tabular coefficients are based 
upon quite smooth interior surfaces, such as occur only in new, 
clean iron pipes, or with fine cement finish, it might be well to 
build the conduit 5 feet or 60 inches in diameter. It is seen 
from the previous example that a semicircular conduit of 6 
feet diameter carries the same amount of water as is here pro- 
vided for. 

A circular conduit running full of water is a long pipe, and 
all the formulas and methods of Arts. 80 and 81 can be applied 
also to their discussion. By comparing the formulas of velocity 
for pipes and conduits, 



it is seen that 



= 2 \fe 



in which f is to be taken from Table XVI. Values of c com- 
puted in this manner will not generally agree closely with the 
coefficients of SMITH, partly because the values of / are given 



Art. g6 ] CIRCULAR CONDUITS, PARTLY FULL. 221 

only to three decimal places, and partly because Table XVI 
was constructed by regarding other discussions. An agree- 
ment within 5 per cent in mean velocities deduced by different 
methods is all that can generally be expected in conduit com- 
putations, and if the actual discharge agrees as closely as this 
with the computed discharge, the designer can be considered 
as a fortunate man. 

All of the laws deduced in the last chapter regarding the 
relation between diameter and discharge, relative discharging 
capacity, etc., hence apply equally well to circular conduits 
which run either full or half full. And if the conduit be full 
it matters not whether it be laid truly to grade or whether a 
portion of it be under pressure, since in either case the slope 
s is the total fall h divided by the total length. Usually, how- 
ever, the word conduit implies a uniform slope for consider- 
able distances, and in this case the hydraulic gradient coincides 
with the surface of the flowing water. 

Prob. 119. Find the discharge of a conduit when running 
full, its diameter being 6 feet and its fall 9.54 feet in one mile. 

Prob. 120. Find the diameter of a conduit to deliver when 
running full 16 500000 gallons per day, its slope being 0.00016. 

Article 96. Circular Conduits, Partly Full. 

Let a circular conduit with the slope s be partly full of 
water, its cross-section being a and hydraulic radius r. Then 
the mean velocity of flow is 

v — c Vrs, 
and the discharge per second is 

q = av = c. a Vrs. 
The mean velocity is hence proportional to Vr and the dis- 
charge to a Vr, provided that ^ be a constant. Since, how- 
ever, c varies slightly with r, this law of proportionality is 
approximate. 



222 



FLOW IN CONDUITS AND CANALS. [Chap. VIII. 



When a circular conduit of diameter d runs either full or 
half full its hydraulic radius is \d (Art. 93). If it is filled to 
the depth d', the wetted perimeter is 

p = ind-\- d arc sin — , 

A* d 




and the sectional area of the water surface is 



Fxo. 67. a = \ d P + \d' ~ ¥) Yd'(d - d'\ 

From these/ and a can be computed, and then r is found by 
dividing a by p. The following table gives values of /, a, and 
r for a circle whose diameter is unity for different depths of 
water. To find from it the hydraulic radius for any other cir- 

TABLE XVIII. CROSS-SECTIONS OF CIRCULAR CONDUITS. 



Depth 


Wetted 
Perimeter 


Sectional Area 


Hydraulic 
Radius 


Velocity 


Discharge 


a 


P 


a 


r 


y r 


a \' r 


Full 1 . 


3-142 


O.7854 


O.25 


o-5 


0-393 


o.95 


2.691 


O.7708 


O.2S6 


0-535 


•413 


0.9 


2.498 


0-7445 


O.29S 


0.546 


.406 


0.81 


2.240 


O.6815 


O.3043 


o-552 


•376 


0.8 


2.214 


O.6735 


O.3042 


0.552 


•372 


0.7 


I.983 


0.5S74 


O.296 


0-544 


.320 


0.6 


1.772 


O.4920 


O.278 


0.527 


•259 


Half Full 0.5 


I- 571 


O.3927 


O.25 


0.5 


.196 


0.4 


I.369 


O.2934 


O.214 


0.463 


.136 


0.3 


I- 159 


O.I9S1 


O.I71 


0.414 


.OS20 


0.2 


O.927 


0.111S 


O.I2I 


0.34S 


.O3S9 


O.I 


O.643 


. 040S 


O.0635 


0.252 


.OIO3 


Empty 0.0 


O.O 


0.0 


O.O 


0.0 


O.O 



cle it is only necessary to multiply the tabular values of r by 
the given diameter d. The table shows that the greatest value 
of the hydraulic radius occurs when d' = o.Sid, and that it is 
but little less when d' = o.Sd. 



Art. g6.] CIRCULAR CONDUITS, PARTLY FULL. 223 

In the fifth and sixth columns of the table are given values 
of Vr and a Vr for different depths in the circle whose diame- 
ter is unity ; these are approximately proportional to the 
velocity and discharge which occur at those depths in a circle 
of any size. The table shows that the greatest velocity occurs 
when the depth of the water is about eight-tenths of the di- 
ameter, and that the greatest discharge occurs when the depth 
is about 0.95^/, or ^-g-ths of the diameter. 

By the help of the above table the velocity and discharge 
may be computed when c is known, but it is not possible on 
account of the lack of experimental knowledge to state precise 
values of c for different values of r in circles of different sizes. 
However, it is known that an increase in r increases c, and that 
a decrease in r decreases c. The following experiments of 
Darcy and Bazin show the extent of this variation for a semi- 
circular conduit of 4.1 feet diameter, and they also teach that 
the nature of the interior surface greatly influences the values 
of c. Two conduits were built each with a slope s = 0.0015 
and d = 4. 1 feet. One was lined with neat cement, and the 
other with a mortar made of cement with one-third fine sand. 
The flow was allowed to occur with different depths, and the 
discharges per second were gauged by means of orifices; this 
enabled the velocities to be computed, and from these the 
values of c were found. The following are a portion of the re- 
sults obtained, d' denoting the depth of water in the conduit, 
and all dimensions being in feet : * 





For cemen 


. lining 






For mortar 


lining 




d' 


r 


V 


c 


d' 


r 


V 


c 


2.05 


1.029 


6.06 


154 


2.04 


1 .022 


5.55 


142 


1.83 


0.949 


5 


75 


152 


1.80 


0.941 


5.20 


138 


1. 61 


0.867 


5 


29 


147 


1.69 


0.900 


4.94 


135 


i-34 


0.750 


4 


87 


145 


1. 41 


0.787 


4-51 


131 


1.03 


0.605 


4 


16 


138 


1 .09 


0.635 


3.87 


125 


0.83 


0.503 


3 


72 


136 


0.88 


0.529 


3-43 


122 


o.59 


0.366 


3 


02 


129 


0.61 


o.379 


2.87 


I20 



* Smith's Hydraulics, p. 176. 



224 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

It is here seen that c decreases quite uniformly with r, and 
that the velocities for the mortar lining are 8 or 10 per cent 
less than for the neat cement lining. 

The value of the coefficient c for these experiments may be 
roughly expressed by the formula 

c — c' — i6(±d— d'\ 

in which c' is the coefficient for the conduit when running half 
full. How this will apply to different diameters and velocities 
is not known; when d' is greater than o.Zd it will probably 
prove incorrect. In practice, however, computations on the 
flow in partly filled conduits are of rare occurrence. 

Prob. 121. Compute the hydraulic radius for a circular 
conduit when it is three-fourths filled with water, and also the 
mean velocity if it be lined with pure cement and laid on a 
grade of 0.15 per 100, the diameter being 4.1 feet. 

Article 97. Open Rectangular Conduits. 

In designing an open rectangular trough or conduit to 
carry water there is a certain ratio of breadth to depth which 
is most advantageous, because that thereby either the dis- 
charge is the greatest or the least amount of material is re- 
quired for its construction. This advantageous proportion is 
the one which offers the least frictional resistance to the flow ; 
in a very wide and shallow trough the friction would be great, 
and the same would be the case in one of small width and 
large depth. It is now to be shown that the least friction, 
and hence the best proportions, results when the width is 
double of the depth. 

The head lost in friction is directly proportional to the 
wetted perimeter and inversely proportional to the area of the 
water cross-section (Art. 94). In order that this may be the 



Art. 97-] OPEN RECTANGULAR CONDUITS. 225 

least possible, the wetted perimeter should be a minimum for 
a given area, or the area should be a maximum for a given 
wetted perimeter. But the ratio of the area to the perimeter 
is the hydraulic radius 

a 

P 

which therefore is to be a maximum, subject to the other con- 
ditions of the problem, in order to secure the most advantage- 
ous cross-section. This is an approximate general rule, appli- 
cable to all kinds of channels, and it is plain that the circle 
fulfils the requirement in a higher degree than any other 
figure. 

For an open rectangular conduit of breadth b and depth d 
the value of the hydraulic radius is 

bd 



b + 2d' 



If it be required to find the most advantageous section for a 
given wetted perimeter, this may be written 

and this is seen to be a maximum when b = \p, that is, when 
b = 2d, or the breadth is double the depth. If, however, it be 
required to determine the most advantageous section for a 
given area, the value of the hydraulic radius may be written 



H^ 



and by equating the first derivative to zero, there is found 
b 2 — 2a, from which b 1 = 2bd, or b = 2d, as before. 



226 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

Again, if it be required to find the most advantageous sec- 
tion to carry q cubic feet of water per second, the hydraulic 
radius 

_ bd 

T ~ t + 2d 

is to be made a maximum, subject to the condition 



b 3 d> 
c . a Vrs = cs? 



b 4- 2d 



Regarding c as a constant, the values of b and d which render 
r a maximum can be ascertained by the rules of the higher 
analysis, and there is also found for this case the relation b = 2d, 
or the breadth is double the depth. 

The velocity and discharge through a rectangular conduit 
are expressed by the general equations 

v = c \ rs, q = av, 

and are computed without difficulty for any given case when 
the coefficient c is known. To ascertain this, however, is not 
easy on account of the lack of experiments by which alone its 
value can be ascertained. When the depth of the water in the 
conduit is one-half of its width, thus giving the most advan- 
tageous section, the values of c for smooth interior surfaces 
may be estimated from the table in Art. 96 for circular con- 
duits, although c is probably smaller for rectangles than for 
circles of equal area. When the depth of the water is less or 
greater than \d, it must be remembered that c increases with r. 
The value of c also is subject to slight variations with the slope 
s, and to great variations with the degree of roughness of the 
surface. 

The following table, derived from Smith's discussion of the 
experiments of Darcy and BAZIN, gives values of c for a num- 



Art. 98.] 



TRAPEZOIDAL SECTIONS. 



227 



ber of wooden and masonry conduits with rectangular sections, 
all of which were laid on the grade of 0.49 feet per 100, or 

TABLE XIX. COEFFICIENTS FOR RECTANGULAR CONDUITS. 



Unplaned Plank. 
b = 3.93 feet. 


Unplaned Plank. 
b = 6.53 feet. 


Pure Cement. 
b = 5.94 feet. 


Brick. 
b = 6.27 feet. 


d 


c 


d 


c 


d 


c 


d 


c 


O.27 


99 


0.20 


89 


O.I8 


116 


0.20 


89 


.41 


108 


•30 


IOI 


.28 


125 


•31 


98 


.67 


112 


.46 


IO9 


•43 


132 


.49 


104 


.89 


114 


.60 


113 


.56 


135 


•57 


105 


I. OO 


114 


.72 


Il6 


.63 


136 


.65 


104 


1. 19 


116 


.78 


Il6 


.69 


136 


•71 


106 


I.29 


117 


.89 


Il8 


.80 


137 


.8.5 


107 


I.46 


118 


•94 


120 


.91 


138 


•97 


no 



s = 0.0049. The great influence of roughness of surface in 
diminishing the coefficient is here plainly seen. For masonry 
conduits with hammer-dressed surfaces c may be as low as 60 
or 50, particularly when covered with moss and slime. 

Prob. 122. Compare the discharge of a trough 1 X 3 feet 
with that of two troughs each 1 X 2 feet. 

Prob. 123. Find the size of a trough, whose width is double 
its depth, which will deliver 125 cubic feet per minute when its 
slope is 0.002, taking c as 100. 

Article 98. Trapezoidal Sections. 

Ditches and conduits are often built with a bottom nearly 
flat and with side slopes, thus forming a trapezoidal section. 
The side slope is fixed by the nature of the soil or by other 
circumstances, the grade is given, and it may be then required 
to ascertain the relation between the bottom width and the 
depth of water, in order that the section shall be the most ad- 
vantageous. This can be done by the same reasoning as used 




228 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

for the rectangle in the last article, but it may be well to em- 
ploy a different method, and thus be able to consider the sub- 
ject in a new light. 

Let the trapezoidal channel have the bottom width b, the 
depth d, and let 6 be the angle made by the side slopes with the 

horizontal. Let it be required to 
discharge q cubic feet per second ; 

iW then 

Fig. 68. q = CO. VrS. 

Now the most advantageous proportions may be said to be 
those that will render the cross-section a a minimum, for thus 
the least excavation will be required. The above equation may 
be written 

q % a % 

In this/ is to be replaced by its value in terms of a and d, and 
then the value of d is to be found which renders a a minimum. 
For this purpose the figure gives 

a = d(b -\-d cot 0) ; 

p = b-\- :-^-5 = -7 + d f-^-2 — cot 0); 

r sin 6 d ' \sin J 

from which the equation becomes 

^ + ? V (si^- COt ")=*"'• 

Obtaining the first derivative of a with respect to d, and equat- 
ing it to zero, there is found 

(-A-, - cot 0)d 2 = a ; 
\sm 6 I 

and replacing for a its value, there results 



d[-^—-2cote ; ( 7 i> 

\sm 



Art. 98.] TRAPEZOIDAL SECTIONS. 229 

which is the relation that gives the most advantageous cross- 
section. If 6 = 90 , the trapezoid becomes a rectangle, and 
b — 2d, as previously deduced. As c has been regarded as a 
constant in this investigation, the conclusion is not a rigorous 
one, although it may be safely followed in practice. It is to be 
expected, as in all cases of a maximum, that quite considerable 
variations in the ratio b : d may occur without materially affect- 
ing the value of a. 

When the value of c is known, the general formulas^ = c Vrs 
and q = av may be used to obtain a rough approximation to 
the discharge. The formula of KUTTER (Art. 101) may be used 
to determine c when the nature of the bed of the channel is 
known. In any important case, however, computations cannot 
be trusted to give reliable values of the discharge on account 
of the uncertainty regarding the coefficient, and an actual 
gauging of the flow should be made. This is best effected by 
a weir, but if that should prove too expensive, the methods 
explained in Chap. IX may be employed to give more precise 
results than can usually be determined by any computation. 

The problem of determining the size of a trapezoidal 
channel to carry a given quantity of water, does not require c 
to be determined so closely. For this purpose the following 
values may be used, the lower ones being for small cross-sec- 
tions with rough and foul surfaces, and the higher ones for 
quite smooth surfaces : 



For unplaned plank, 


c = 100 to 120 


For smooth masonry, 


c = 90 to no 


For clean earth, 


c — 60 to 80 


For stony earth, 


c = 40 to 60 


For rough stone, 


c= 35 to 50 



For earth foul with weeds, c = 30 to 50 



230 FLOW IX COXDUITS AXD CAXALS. [Chap. VIII. 

To solve this problem, let a and p be replaced by their values 
in terms of b and d. The discharge then is 



,,, , , n /d(b 4- d cot 6)s sin 6 
V &_sm0 -\-2d 

Now when £, r. 0, and 5 are known, the equation contains two 
unknown quantities, and af. If the section is to be the most 

advantageous,, b can be replaced by its value in terms of d as 
above found, and the equation then has but one unknown. Or 
in general, if b = md, where m is any assumed number, the 
solution gives 

d - 0= g\m sin fl + 2) 

c's\j)i — cot 6f sin 6' 

For the particular case where the side slopes are 1 to I or 
6 — 45°, and the bottom width is to be equal to the water 
depth, or m — I, this becomes 



«)' 



d — o. 

Vs. 

These formulas are analogous to those for finding the diameter 
of pipes and circular conduits, and the numerical operations 
are in all respects similar. It is plain that by assigning dif- 
ferent values to m numerous sections may be determined 
which will satisfy the imposed conditions, and usually the one 
is to be selected that will give both a safe velocity and a 
minimum cost. In Art. 103 will be found an example of the 
determination of the size of a trapezoidal canal. 

Prob. 124. If the value of c is 71, compute the depth of a 
trapezoidal section to carry 200 cubic feet of water per second, 

6 being 45°, the slope s being 0.001, and the bottom width 
being equal to the depth. Compute also the mean velocity 
for the section. 




Art. 99-] HORSE-SHOE CONDUITS. 23 1 

Article 99. Horse-shoe Conduits. 

In Fig. 69 is given an outline cross-section of the Sudbury 
conduit, the flow of which was gauged by FTELEY and 
Stearns, whose discussions have determined a 
formula for its mean velocity. The section 
consists of a part of a circle of 9.0 feet diameter, 
having an invert of 13.22 feet radius, whose 
span is 8.3 feet and depression 0.7 feet, the 
axial depth of the conduit being y.7 feet. The 
conduit is lined with brick, having cement joints one quarter of 
an inch thick. The flow was allowed to occur with different 
depths, for each of which the discharge was determined by 
weir measurement. A discussion of the results led to the 
conclusion that in the portion with the brick lining the coeffi- 
cient c had the value I2yr - 12 when r is in feet, and hence 

v— I2yr - 12 V^rs = J27r°- 62 s°- s (72) 

In a portion of the conduit where the brick lining was coated 
with pure cement the coefficient was found to be from 7 to 8 
per cent greater than 127. In another portion where the brick 
lining was covered with a cement wash laid on with a brush 
the coefficient was from 1 to 3 per cent greater. For a long 
tunnel in which the rock sides were ragged, but with a smooth 
cement floor, it was found to be about 40 per cent less.* 

These results clearly show that the coefficient c increases 
with r, and that it is greatly influenced by the nature of the 
interior surfaces. For sections of smaller area than that above 
given the value of c is undoubtedly less than I27r°- 12 , and for 
those of larger area it is greater ; the extent of variation may 
perhaps be inferred from the table in Art. 95. The general 

* Transactions American Society Civil Engineers 1883, vol. xii. p. 114. 



232 FLOW IN COXDUITS AND CAXALS. [Chap. VIII. 

slope of the Sudbury conduit is about one foot per mile, and c 
is also subject to variation with s, as well as with the tempera- 
ture of the water. Although the above formula is a special 
one, applicable to a single conduit, it is nevertheless of great 
value, as it presents the only existing evidence regarding the 
coefficients for large aqueducts. 

Prob. 125. The actual discharge of the Sudbury conduit is 
about 60 080 000 gallons per 24 hours when the water is 4 feet 
deep, a being 33.31 sq. feet, p =15.21 feet, and s = 0.0001895. 
Compute the discharge by the use of the above formula. 

Article 100. Lampe's Formula. 

The formula given in Art. 91 for the mean velocity of flow 
in long circular pipes can be also applied to conduits with very 
smooth surfaces. Replacing for the ratio h -f- / the slope s, 
and for d its equivalent 4^, it becomes 

v — 203r°- 69 V 555 . ....... (73) 

This formula may be also written 

v = 203r ai % -°55 VrT, (73)' 

in which the quantity preceding the radical in the second mem- 
ber is the coefficient c. According to this empirical expression 
c increases both with r and s, but only slightly with the latter. 
It is probable that this formula represents quite accurately the 
laws of flow in conduits, but the varying degree of roughness 
of surface is not taken into account by it, so that in general it 
can only be used to furnish approximate results, except for the 
case of metal pipes or similar smooth surfaces. For this pur- 
pose the formulas for q and d, given in Art. 91, may be directly 
used for circular sections. It is probable that future researches 
may show that a formula similar to the above may fairly repre- 



Art. ioi.] KUTTER'S FORMULA. .233 

sent all cases, the constant 203 being varied with the roughness 
of the surface. 

Prob. 126. Solve Prob. 125 by the use of Lampe's formula, 
and compare the error of the result with that as deduced by 
the special formula for the conduit. 



Article ioi. Kutter's Formula. 

The researches of GANGUILLET and KUTTER have furnished 
a general expression for the coefficient c in the formula for 
mean velocity, 

v ■= c Vrs, 

by which its value can be computed for any given case when 
the nature of the interior surface is known. This expression is, 
for English measures, 

1.811 „ 0.00281 

r- 41.65+ 

n s , , 

c ~ ~ ~n~i ~ 0.0028 1\' ' ' * * [74) 

in which n is an abstract number whose value depends only 
upon the roughness of the surface, and 

n — 0.009 for well-planed timber ; 

n = 0.010 for neat cement ; 

n — 0.011 for cement with one-third sand; 

n = 0.C12 for unplaned timber ; 

n ±= 0.013 for ashlar and brickwork; 

n = 0.015 for unclean surfaces in sewers and conduits; 

n = 0.017 for rubble masonry ; 

n = 0.020 for canals in very firm gravel ; 



234 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

n = 0.025 for canals and rivers free from stones and weeds; 
71 = 0.030 for canals and rivers with some stones and weeds ; 
n = 0.035 for canals and rivers in bad order. 

By inserting this value of c in the formula for v, the mean 
velocity is made to depend upon r, s, and the roughness of the 
surface. 

The formula of KUTTER has received a wide acceptance on 
account of its application to all kinds of surfaces. Notwith- 
standing that it is purely empirical, and hence not perfect, it is 
to be regarded as a formula of great value, so that no design 
for a conduit or channel should be completed without employ- 
ing it in the investigation, even if the final construction be not 
based upon it. In sewer work it is extensively employed, u 
being taken as about 0.015. The formula shows that c always 
increases with r, that it decreases with s when r is greater than 
3.28 feet, and that it increases with s when r is less than 3.28 

T S T T 

feet. When r equals 3.28 feet the value of c is simply — — . 

It is not likely that future investigations will confirm these laws 
of variation in all respects. 

In the following articles are given values of c for a few 

cases, and these might be greatly extended, as has been done 

by KUTTER and others. But this is scarcely necessary except 

for special lines of investigation, since for single cases there 

is no difficulty in directly computing it for given data. For 

instance, take a rectangular trough of unplaned plank 3.93 

feet wide on a slope of 0.0049, the water being 1.29 feet deep. 

Here 

s — 0.0049 

and 

3.93 X 1.29 f 

r = * 1 „ „q = 0-779 fee t. 



Art. 102.] SEWERS. 235 

Then n being 0.012, the value of c is found to be 

1.811 , ,- . 0.00281 

+ 41.65 -f 

0.012 0.0040 

C — ~ -7 — r = 123. 

, O.OI2 / 0.0028 1 \ 

V 0.779 0.0049/ 

The data here used are taken from the table in Art. 97, where 
the actual value of c is given as 117; hence in this case 
Kutter's formula is about 5 per cent in excess. As a second 
example, the following data from the same table will be taken : 
a rectangular conduit in pure cement, b = 5.94 feet, d = 0.91 
feet, s = 0.0049. Here n = 0.0 10, and r = 0697 feet. Insert- 
ing all values in the formula, there is found c = 148, which is 
8 per cent greater than the true value, 138. Thus is shown the 
fact that errors of 5 and 10 per cent are to be regarded as com- 
mon in calculations on the flow of water in conduits and canals. 

Prob. 127. Compute by Kutter's formula the discharge 
for the data in Prob. 125. 

Article 102. Sewers. 

Sewers smaller in diameter than 18 inches are always cir- 
cular in section. When larger than this they are built with 
the section either circular, egg-shaped, or of the horseshoe 
form. The last shape is very disadvantageous when a small 
quantity of sewage is flowing, for the wetted perimeter is then 
large compared with the area, the hydraulic radius is small, and 
the velocity becomes low, so that a deposit of the foul materials 
results. As the slope of sewer lines is often very slight, it is 
important that such a form of cross-section should be adopted 
to render the velocity of flow sufficient to prevent this deposit. 
A velocity of 2 feet per second is found to be about the mini- 
mum allowable limit, and 4 feet per second need not be usu- 
ally exceeded. 



236 



FLOW IX COXDUITS AXD CAXALS. [Chap. Till. 




The egg-shaped section is designed so that the hydraulic 
radius may not become small even when a small amount of 

sewage is flowing. One of the 
most common forms is that shown 
in Fig. 70, where the greatest width 
DD is two-thirds of the depth 
HM. The arch DHD is a semi- 
circle described from A as a centre. 
The invert LML is a portion of 
a circle described from B as a cen- 
tre, the distance BA being three- 
fourths of DD and the radius BM being one-half of AD. 
Each side DL is described from a centre C so as to be tangent 
to the arch and invert. These relations may be expressed 
more concisely by 

HM = iiD, AB = ID, BM = \D, CL = iW, 

in which D is the horizontal diameter DD. 

Computations on egg-shaped sewers are usually confined to 
three cases, namely, when flowing full, two-thirds full, and one- 
third full. The values of the sectional areas, wetted perimeters, 
and hydraulic radii for these cases, as given by FLYNN,* are 

a p r 

1.1485Z} 2 3.965Z) 

0.7558^ 2 

0.2840/)' 2 1 



Full 

Two-thirds full 
One-third full 



m D 
\7& 



0.2S9JD 
0.3157Z} 
0.2066D 



This shows that the hydraulic radius, and hence the velocity, 

is but litt 

with full section. 



is but little less when flowing one-third full than when flowing 



E gg- 



-shaped sewers and small circular ones are formed by 
laying consecutive lengths of clay or cement pipe whose interior 



Van Xostrand's Magazine. 1SS3. vol. xxviii. p. 138. 



Art 102.] SEWERS. 2^J 

surfaces are quite smooth when new, but may become foul after 
use. Large sewers of circular section are made of brick, and 
are more apt to become foul than smaller ones. In the separate 
system, where systematic flushing is employed and the pipes 
are small, foulness of surface is not so common as in the com- 
bined system, where the storm water is alone used for this 
purpose. In the latter case the sizes are computed for the 
volume of storm water to be discharged, the amount of sewage 
being very small in comparison. 

The discharge of a sewer pipe enters it at intervals along 
its length, and hence the flow is not uniform. The depth of 
the flow increases along the length, and at junctions the size of 
the pipe is enlarged. The strict investigation of the problem 
of flow is accordingly one of great complexity. But consider- 
ing the fact that the sewer is rarely filled, and that it should be 
made large enough to provide for contingencies and future 
extensions, it appears that great precision is unnecessary. The 
universal practice, therefore, is to discuss a sewer for the con- 
dition of maximum discharge, regarding it as a channel with 
uniform flow. The main problem is that of the determination 
of size ; if the form be circular, the diameter is found, as in 
Art. 95, by 

W Vsl v V* 

If the form be egg-shaped and of the proportions above ex- 
plained, the discharge when running full is 



q = acVrs = 1. 1 48 5 D Wo. 2 897DS, 
from which the value of D is found to be 

.1 

D = 1.21 



Vs 



Thus when q has been determined and c is known the required 
sizes for given slopes can be computed. The velocity should 



233 



FLO W IX COXDUITS AXD CAXALS. [Chap. VIII. 



also be found in order to ascertain if it be high enough to 
prevent deposit (Art. 108). 

Few or no experiments exist from which to directly deter- 
mine the coefficient c for the flow in sewers, but since the sew- 
age is mostly water, it may be approximately ascertained from 
the values for similar surfaces. KUTTER's formula has been 
extensively employed for this purpose, using 0.015 for the 
coefficient of roughness. The following table gives values of c 
for three different slopes and for two classes of surfaces. The 
values for the degree of roughness represented by n = 0.017 

TABLE XX. COEFFICIENTS FOR SEWERS. 



Hydraulic 
Radius r 
in Feet. 


s = 


00005 


s = c 


.0001 


.$■ = 


0.01 


n = 0.015 


n = 0.017 


n c= 0.015 


n = 0.017 


■n — 0.015 


71 = O.OI7 


0.2 


52 


43 


53 


48 


68 


57 


0-3 


60 


5i 


66 


56 


76 


64 


0.4 


65 


56 


73 


6l 


S3 


70 


0.6 


76 


65 


82 


70 


90 


76 


O.S 


82 


72 


37 


76 


95 


82 


I. 


88 


77 


92 


So 


99 


37 


i-5 


100 


86 


103 


S 9 


10S 


93 


2. 


106 


94 


10S 


96 


in 


99 


3- 


116 


103 


11S 


104 


11S 


105 

\ 



are applicable to sewers with quite rough surfaces of masonry ; 
those for n = 0.015 are applicable to sewers with ordinary 
smooth surfaces, somewhat fouled or tuberculated by deposits, 
and are the ones to be generally used in computations. By 
the help of this table and the general equations for mean 
velocity and discharge all problems relating to flow in sewers 
can be readily solved. 

Prob. 128. The grade of a sewer is one foot in 960, and its 
discharge is to be 65 cubic feet per second. What is the diam- 
eter of the sewer if circular ? 



Art. 103.] 



DITCHES AND CANALS. 



239 



Article 103. Ditches and Canals. 

Ditches for irrigating purposes are of a trapezoidal section, 
and the slope is determined by the fall between the point 
from which the water is taken and the place of delivery. If 
the fall is large it may not be possible to construct the ditch in 
a straight line between the two points, even if the topography 
of the country should permit, on account of the high velocity 
which would result. A velocity exceeding 2 feet per second 
may often prove injurious in wearing the bed of the channel 
unless protected by riprap or other lining. For this reason as 
well as for others the alignment of ditches and canals is often 
circuitous. 



The principles of the preceding articles are sufficient to 

solve all usual problems of uniform flow in such channels when 

the values of c are known. These are perhaps best determined 

by KUTTER's formula, and for greater convenience a table is 

TABLE XXI. COEFFICIENTS FOR CHANNELS IN EARTH. 



Hydraulic 
Radius r 
in Feet. 


s = 


00005 


.y = 0.0001 


j = 


O.OI 


n = 0.025 


n = 0.030 


n~ 0.025 


n — 0.030 


n = 0.025 


n — 0.030 


O 


5 


33 


31 


41 


33 


47 


37 


I 




49 


40 


52 


42 


56 


45 


I 


5 


57 


47 


59 


48 


62 


5i 


2 




64 


52 


65 


53 


67 


54 


3 




72 


59 


72 


59 


72 


60 


4 




77 


64 


77 


64 


76 


63 


5 




Si 


68 


80 


68 


79 


66 


6 




86 


72 


84 


7i 


So 


68 


8 




9i 


76 


87 


74 


82 


70 


10 




96 


80 


9i 


80 


85 


73 


15 




105 


89 


97 


84 


90 


77 


25 




114 


100 


IOI 


92 


95 


82 



240 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

here given, showing their average values for three slopes and 
two degrees of roughness. 

As an example of the use of the table let it be required to 
find the width and depth of a ditch of most advantageous cross- 
section, whose channel is to be in tolerably good order, so that 
n = 0.025. The amount of water to be delivered is 200 cubic 
feet per second and the grade is 1 in 1000, the side slopes of 
the channel being 1 to 1. From Art. 98 the relation between 
the bottom width and the depth of the water is, since 6 = 45 , 

— 2 cot 0) = 0.828;/. 



.sin & 
The area of the cross-section then is 

a = d(&4-d cot 6) = 1.828a 72 , 
and the wetted perimeter is 

2d 

r ' ship J J ' 

whence the hydraulic radius is found to be 
1.828a 72 



3.656a 7 

It is indeed a general rule, which might properly have been 
set forth in Art. 98, that the hydraulic radius is one-half the 
depth of the water in trapezoidal channels of most advanta- 
geous cross-section. Now, since d is unknown c cannot be 
taken from the table, and as a first approximation let it be sup- 
posed to be 60. Then in the general formula for discharge the 
above values are substituted, giving 



200 = 60 x 1.828a 7 - \ 0.5^ x 0.001, 

from which d is found to be 5.8 feet. Accordingly r = 2.9 
feet, and from the table c is about 71. Repeating the compu- 
tation with this value of c there is found d— 5.44 feet, which, 



Art. 103.] DITCHES AND CANALS. 24 1 

considering the uncertainty of c, is sufficiently close. The 
depth may then be made 5.5 feet, and the bottom width will 

be 

b = 0.828 X 5-5 = 4-55 feet, 

and the sectional area is 

a — 1.828 x 5»5 2 = 55.3 square feet, 

which gives for the velocity 

200 . . 

v = = 1.62 feet per second. 

55-3 3 P 

This completes the investigation if the velocity is regarded as 
satisfactory. But for most earths this would be too high, and 
accordingly the section must be made wider and of less depth 
in order to reduce the hydraulic radius and diminish the ve- 
locity. 

The following statements show approximately the veloci- 
ties which are required to move different materials : 

0.25 feet per second moves fine clay, 

0.5 feet per second moves loam and earth, 

1.0 feet per second moves sand, 

2.0 feet per second moves gravel, 

3.0 feet per second moves pebbles 1 inch in size, 

4.0 feet per second moves spalls and stones, 

6.0 feet per second moves large stones. 

The mean velocity in a channel may be somewhat larger than 
these values before the materials will move, because the veloci- 
ties along the wetted perimeter are smaller than the mean 
velocity. More will be found on this subject in Art. 107. 

Prob. 129. Compute the mean velocity in a ditch which is 
to discharge 200 cubic feet per second on a grade of 1 in 1000 
when its bottom width is 16 feet and the side slopes are 1 to I. 
Ans. d = 3.09 feet, v = 3.4 feet, per second. 




242 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

Art. 104. Losses of Head. 

The only loss of head thus far considered is that due to 
friction, but other sources of loss may often exist. As in the 
flow in pipes, these may be classified as losses at entrance, losses 
due to curvature, and losses caused by obstructions in the chan- 
nel or by changes in the area of cross-section. 

When water is admitted to a channel from a reservoir or 
pond through a rectangular sluice there occurs a contraction 

similar to that at the entrance 
into a pipe, and which may be 
often observed in a slight de- 
pression of the surface, as at D 
in the diagram. At this point, 
therefore, the velocity is greater than the mean velocity v, and 
a loss of energy or head results from the subsequent expansion, 
which is approximately measured by the difference of the 
depths d x and d„, the former being taken at the entrance of the 
channel, and the latter below the depression where; the uniform 
flow is fully established. According to the experiments of 
DUBUAT, the loss of head is measured by 

v 2 

d, — d„ — m — , 

ocr 

in which m ranges between o and 2 according to the condition 
of the entrance. If the channel be small compared with the 
reservoir, and both the bottom and side edges of the entrance 
be square, m may be nearly 2 : but if these edges be rounded, m 
may be very small, particularly if the bottom contraction is 
suppressed. All the remarks in Chapter IV relating to sup- 
pression of the contraction apply here, and in a short channel 
or flume it may be important to prevent this loss of head by a 
rounded or curved approach. 



Art. 104.] LOSSES OF HEAD. 243 

The loss of head due to bends or curves in the channel is 
small if the curvature be slight. Undoubtedly every curve 
offers a resistance to the change in direction of the velocity, 
and thus requires an additional head to cause the flow beyond 
that needed to overcome the frictional resistances. Several 
formulas have been proposed to express this loss, but they all 
seem unsatisfactory, and hence will not be presented here, par- 
ticularly as the data for determining their constants are very 
scant. It will be plain that the loss of head due to a curve 
increases with its length and decreases with its radius. Art. 
131 gives a discussion concerning the cause of losses in bends 
and curves. 

The losses of head caused by sudden enlargement or by 
sudden contraction of the cross-section of a channel may be 
estimated by the rules deduced in Arts. 68 and 69. In order 
to avoid these losses changes of section should be made grad- 
ually, so that energy may not be lost in impact. Obstructions 
or submerged dams maybe regarded as causing sudden changes 
of section, and the accompanying losses of head are governed 
by similar laws. The numerical estimation of these losses will 
generally be difficult, but the principles which control them 
will often prove useful in arranging the design of a channel so 
that the maximum work of the water can be rendered avail- 
able. But as all losses of head are directly proportional to the 

velocity-head — , it is plain that they can be rendered inappre- 

ciable by giving to the channel such dimensions as will render 
the mean velocity very small. This may sometimes be impor- 
tant in a short conduit or flume which conveys water from a 
pond or reservoir to a hydraulic motor, particularly in cases 
where the supply is scant, and where all the available head is 
required to be utilized. 

If no losses of head exist except those due to friction the 



244 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

slope s can be computed from (70) if the mean velocity v, the 
hyclraulic radius r, and the coefficient c are known, and then the 
loss of head or fall in a given distance / is 

h —- Is = —> 
c r 

but the numerical value of this will usually be found to be sub- 
ject to a large percentage of error. 

A recent discussion by FOSS furnishes a new series of 
formulas for channels. For surfaces corresponding to Kut- 
TER'S values of n less than 0.017 he finds 

2r&=Cr%s whence h = ^-|, .... (75) 

in which C has the following values: 

For;/ = 0.009, 0.010, 0.011, 0.012, 0.013, 0.015, 0.017 
£7=23000, 19000, 15000, 12000, 10 000, 8000, 6000 

For surfaces corresponding to KUTTER'S values of 11 greater 
than 0.018, his formula is 

v* = Crh whence h = ~^~i, .... {?$') 

and the values of C are 

for n = 0.020, 0.025, 0.030, 0.035 
C— 5000, 3000, 2000, 1500 

For circular pipes running full he also proposes the formula 
d 



s — 0.00C5 — • 



Prob. 130. Compute the fall of the water surface in a length 
of 1000 feet for a ditch where v= 3.62 feet per second, r = 
2.75 feet, and n = 0.025 ; first, by formula (30), and second, by 
FOSS' formula. 

* Journal of Association of Engineering Societies, June, 1894. 



Art, 105.] THE ENERGY OF THE FLOW. 

Article 105. The Energy of the Flow. 



245 



If all the filaments of a stream of water in a pipe or channel 
have the same uniform velocity v, the energy per second is the 
weight, of the discharge multiplied by the velocity-head ; or 



K 



v< v< v> 

W — == zvq — = wa — 



in which a is the area of the cross-section, H^the weight of the 
water delivered per second, and w is the weight of one cubic 
foot. In this case, therefore, the energy of the flow is propor- 
tional to the area of the cross-section and to the cube of the 
velocity. 

The velocities of the filaments in a cross-section are, how- 
ever, not uniform, some being greater and others less than the 
mean velocity v, so that the above expression for K does not 
truly represent the energy of the flow. The experiments of 



nJ 



SCALE OF FEET 




Fig. 7 a. 

FTELEY and STEARNS* on the Sudbury conduit furnish the 
means of comparing the true and mean energies for several cases, 
one of which is represented in Fig. 72. This shows the cross- 
section of the conduit when the water was about 3 feet deep, the 

*Transactions American Society of Civil Engineers, 1883, vol. xii. p. 324. 



246 FLOW IN CONDUITS AND CANALS. [Chap. VIII. 

97 dots being the points at which the velocities were measured 
by a current meter (Art. 1 09). As these points are distributed 
over the cross-section with tolerable uniformity it may be con- 
sidered as divided into 97 equal parts, in each of which the 
velocity is that observed. The true energy of the flow, then, is 

97 \2£ 2g 2gJ 

and the ratio of the true energy to the mean energy is 
K' 97 r' 



A c x -J- C Q 



3 
97 



For this case the mean velocity v is the average of the 97 ob- 
servations, or v = 2.620 feet per second. Cubing this and also 
each individual velocity, and inserting in the formula, there is 
found K' = 0.9992K, so that in this instance the two energies 
are practically equal. 

Prob. 131. The area of the cross-section of the water in Fig. 
72 being 2.458 square feet, compute the theoretic energy per 
second, and the theoretic horse-power. 



Art. 106.] BROOKS AND RIVERS. 247 



CHAPTER IX. 
FLOW IN RIVERS. 

Article 106. Brooks and Rivers. 

No branch of Hydraulics has received more detailed investi- 
gation than that of the flow in river channels, and yet the sub- 
ject is but imperfectly understood. The great object of all 
these investigations has been to devise a simple method of de- 
termining the mean velocity and discharge without the neces- 
sity of expensive field operations. In general it may be said 
that this end has not yet been attained, even for the case of 
uniform flow. Of the various formulas proposed to represent 
the relation of mean velocity to the hydraulic radius and the 
slope, none have proved to be of general practical value except 
the empirical expression used in the last chapter, and this is 
often inapplicable on account of the difficulty of measuring s 
and determining c. The fundamental equations for discussing 
the laws of variation in the mean velocity v and in the dis- 
charge q are 

v = c V rs, q = a . c Vrs ; 

and all the general principles of the last chapter are to be taken 
as directly applicable to uniform flow in natural channels. 

KUTTER'S formula for the value of c is probably the best 
in the present state of science, although it is now generally 
recognized that it gives too large values for small slopes. In 
using it the coefficients for rivers in good condition may be 
taken from Art. 103, but for bad regimen n is to be taken at 0.03, 
and for wild torrents at 0.04 or 0.05. It is, however, too much to 



24.3 FLOW IN RIVERS. [Chap. IX. 

expect that a single formula should accurately express the 
mean velocity in small brooks and large rivers, and the general 
opinion now is that efforts to establish such an expression will 
not prove successful. In the present state of the science no 
engineer can afford in any case of importance to rel)~ upon a 
formula to furnish anything more than a rough approximation 
to the discharge in river channels, but actual field measure- 
ments of velocity must be made. 

When the above formulas are used to determine the dis- 
charge of a river a long straight portion or reach should be 
selected, where the cross-sections are uniform in shape and size. 
The width of the stream is then divided into a number of parts, 
and soundings taken at each point of division. The data are 
thus obtained for computing the area a and the wetted perime- 
ter/, from which the hydraulic depth r is derived. To deter- 
mine the slope s a length / is to be measured, at each end of 
which bench marks are established whose difference of elevation 
is found by precise levels. The elevations of the water surfaces 
below these benches are then to be simultaneously taken, 
whence the fall h in the distance / becomes known. As this 
fall is often small, it is very important that every precaution be 
taken to avoid error in the measurements, and that a number 
of them be taken in order to secure a precise mean. Care 
should be observed that the stage of water is not varying while 
these observations are being made, and for this and other pur- 
poses a permanent gauge must be established. It is also very 
important that the points upon the water surface which are 
selected for comparison should be situated so as to be free from 
local influences such as eddies, since these often cause marked 
deviations from the normal surface of the stream. If hook 
gauges can be used for referring the water levels to the benches 
probably the most accurate results can be obtained. It has 
been observed that the surface of a swiftly flowing stream is 



Art. 107.] VELOCITIES IN A CROSS-SECTION. 249 

not a plane, but a cylinder, which is concave to the bed, its 
highest elevation being where the velocity is greatest, and 
hence the two points of reference should be located similarly 
with respect to the axis of the current. In spite of all precau- 
tions, however, the relative error in h will usually be large in 
the case of slight slopes, unless / be very long, which cannot 
often occur in streams under conditions of uniformity. 

Owing to the uncertainty of determinations of discharge 
made in the manner just described, the common practice is to 
gauge the stream by velocity observations, to which subject 
therefore a large part of this chapter will be devoted. The 
methods given are equally applicable to conduits and canals, 
and in Art. 115 will be found a summary which briefly com- 
pares the various processes. 

Prob. 132. Which has the greater discharge — a stream 2 
feet deep and 85 feet wide on a slope of 1 foot per mile, or a 
stream 3 feet deep and 40 feet wide on a slope of 2 feet per 
mile ? 



Article 107. Velocities in a Cross-section. 

The mean velocity v is the average of all the velocities of 
all the small sections or filaments in a cross-section (Art. 93). 
Some of these individual velocities are much smaller, and oth- 
ers materially larger, than the mean velocity. Along the bot- 
tom of the stream, where the frictional resistances are the great- 
est, the velocities are the least ; along the centre of the stream 
they are the greatest. A brief statement of the general laws 
of variation of these velocities is now to be made. 

In Fig 73 there is shown at A a cross-section of a stream 
with contour curves of equal velocity ; here the greatest veloc- 
ity is seen to be near the deepest part of the section a short 



250 



FLOW IN RIVERS. 



[Chap. IX. 



distance below the surface. At B is shown a plan of the 
stream with arrows roughly representing the intensities of the 
surface velocities at different points ; the greatest of these is 
seen to be near the deepest part or axis of the channel while 
the others diminish toward the banks, the law of variation be- 
ing a curve resembling a parabola. At C is shown by arrows 
the variation of velocities in a vertical line, the smallest being 




Fig. 73. 

at the bottom, and the largest a short distance below the sur- 
face ; concerning this curve there has been much contention, 
but it is commonly thought to be a parabola whose axis is hori- 
zontal. These are the general laws of the variation of velocity 
throughout the cross-section ; the particular relations are of a 
complex character, and vary so greatly in channels of different 
kinds that it is difficult to formulate them, although many at- 
tempts to do so have been made. Some of these formulas 
which connect the mean velocity with particular velocities, such 
as the maximum surface velocity, mid depth velocity in the 
axis of the stream, etc., will be given in the following articles 
in connection with the subject of gauging rivers. 



In a straight channel whose bed is of a uniform nature the 
deepest part is near the middle of its width, while the two sides 
are approximately symmetrical. In a river bend, however, 
the deepest part is near the outer bank, while on the inner 
side the water is shallow : the cause of this is undoubtedly due 
to the centrifugal force of the current, which, resisting the 



Art. 108.] TRANSPORTING CAPACITY OF CURRENTS. 25 1 

change in direction, creates currents which scour away the 
outer bank or prevents deposits from there occurring. It is 
well known to all, that rivers of the least slope have the most 
bends ; perhaps this is due to the greater relative influence of 
such cross currents (Art. 131). 

The theory of the flow of water in channels, like that of 
flow in pipes, is based upon the supposition of a mean velocity 
which is the average of all the parallel individual velocities in 
the cross-section. But in fact there are numerous sinuous 
motions of particles from the bottom to the surface which also 
consume a portion of the lost head. The influence of these 
sinuosities is as yet but little understood; when in the future 
this becomes known a better theory may be possible. 

Prob. 133. Find the approximate discharge of a stream 
whose width is 200 feet, depth 3 feet, slope 0.6 feet per mile, 
when the bottom is very stony and in bad condition. 

Article 108. The Transporting Capacity of Currents. 

The fact that the water of streams transports large quan- 
tities of earthy matter, either in suspension or by rolling it 
along the bed of the channel, is well known, and has already 
been mentioned in Article 103. It is now to be shown that 
the diameters of bodies which can be moved by the pressure 
of a current vary as the square of its velocity, and their weights 
vary as the sixth power of the velocity. 

When water causes sand or pebbles to roll along the bed 
of a channel it must exert a force approximately proportional 
to the square of the velocity and to the area exposed (Art. 32), 
or if d be the diameter of the body and a a constant, 

F = ad 2 v\ 
But if motion just occurs, this force is also proportional to the 
weight of the body, because the frictional resistances of one 



252 FLOW IN RIVERS. [Chap. IX. 

body upon another varies as the normal pressure or weight. 
And as the weight varies as the cube of the diameter, 



d s = ad 2 v\ or d 



av . 



Now since d varies as v\ the weight of the body, which is pro- 
portional to d 3 , must vary as v* ; which proves the proposition 
as enunciated. 

Since the weight of sand and stones when immersed in 
water is only about one-half their weight in air, the frictional 
resistances to their motion are slight, and this helps to explain 
the circumstance that they are so easily transported by currents 
of moderate velocity. It is found by observation that a pebble 
about one inch in diameter is rolled along the bed of a channel 
when the velocity is about 3-5- feet per second ; hence, according 
to the above theoretical deduction, a velocity 5 times as great, 
or 17-J- feet per second, will carry along stones of 25 inches 
diameter. This law of the transporting capacity of flowing 
water is only an approximate one, for the recorded experiments 
seem to indicate that the diameters of moving pebbles on the 
bed of a channel do not vary quite as rapidly as the square of 
the velocity. The law, moreover, is applicable only to bodies 
of similar shape, and cannot be used for comparing round peb- 
bles with flat spalls. 

The following table gives the velocities on the bed or bot- 
tom of the channel which are required to move the materials 
stated. The corresponding approximate mean velocities in 
the cross-section given in the last column are derived from the 
empirical formula deduced by Darcy, 

v = v' -\- 11 \ rs, 

in which v' is the bottom and v the mean velocity. The bot- 
tom or transporting velocities were deduced by DUBUAT from 
experiments in small troughs, and hence are probably slightlj 



Art. rog.] THE CURRENT METER. 2$$ 

less than the velocities which would move the same mate- 
rials in channels of natural earth. 



Bottom 


Mean 


velocity. 


velocity 


0-3 


O.4 


O.4 


O.5 


O.6 


O.8 


1.2 


1.6 


2-5 
3-5 


3-5 
4.5 



Clay, fit for pottery, 

Sand, size of anise-seed, 

Gravel, size of peas, 

Gravel, size of beans, 

Shingle, about 1 inch in diameter, . . . 
Angular stones, about i-J inches diameter, 

The general conclusion to be derived from these figures is 
that ordinary small, loose earthy materials will be transported 
or rolled along the bed of a channel by velocities of 2 or 3 feet 
per second. It is not necessarily to be inferred that this 
movement of the materials is of an injurious nature in streams 
with a fixed regimen, but in artificial canals the subject is one 
that demands close attention. The velocity of the moving 
objects after starting has been found to be usually less than 
half that of the current.* 

Prob. 134. A stone weighing 0.5 pounds is moved by a 
current of 3 feet per second ; what weight will be moved by a 
current of 9 feet per second ? 

Article 109. The Current Meter. 

The most convenient way of precisely measuring the dis- 
charge of a canal, conduit, or small stream is by means of a 
weir which is specially built for that purpose. The flow of a 
very large conduit or of a large stream cannot, however, be 
successfully gauged in this manner, both on account of the 
expense of the dam and weir, and because the weir coefficients 
are not well known for depths of water greater than about 1.5 

* See paper by Herschel on the erosive and abrading power of water, in 
Journal of the Franklin Institute, May, 1878. 



254 FLOW IX RIVERS. [Chap. IX. 

feet. Large quantities of water, therefore, are usually meas- 
ured by observing the velocity of its flow, and the current 
meter furnishes a method of doing this which is extensively 
used, and which gives accurate results. 

The current meter is like a windmill, having three or more 
vanes mounted on a spindle, and so arranged that the face of 
the mill or wheel always stands normal to the current, the 
pressure of which causes it to revolve. The number of revo- 
lutions of the wheel is approximately proportional to the ve- 
locity of the current. In the best forms of instruments the 
number of revolutions made in a given time is determined by 
an apparatus on shore or in a boat from which wires lead to 
the meter under water ; at every revolution an electric connec- 
tion is made and broken which affects a dial on the recording 
apparatus. The observer has hence only to note the time of 
beginning and ending of the experiment, and to read the num- 
ber of revolutions which have occurred during the interval. 
For a canal or small stream the meter is best operated from a 
bridge ; in large streams a boat must be used. 

To derive the velocity from the number of recorded revo- 
lutions per second, the meter must be first rated by pushing it 
at a known velocity in still water. For this purpose a base 
line several hundred feet long is laid out on shore, and ranges 
established so that a boat may be rowed over the same dis- 
tance and the time of its passage be determined. The current 
meter is placed in the bow of the boat, and a start made suffi- 
ciently far from the base so that a uniform velocity can be 
acquired before reaching it : the distance is then traversed 
with this uniform velocity and the times observed, as also the 
actual records of the meter. It is usually found that the num- 
ber of revolutions are not exactly proporcional to the a( 
velocities of the boat, and hence it is necessary to run the boat 



Art. 109.] THE CURRENT METER. 255 

at different velocities per second and ascertain the correspond- 
ing number of revolutions of the wheel for each. A table may 
then be prepared which gives the velocity corresponding to 
the revolutions per second, from which in subsequent field 
work the reductions can readily be made. The relation be- 
tween the velocity V and the number of revolutions per 
second n can also be expressed by an equation of the form 

V = a -\- fin -j- yri 1 , 

and the experiments furnish the data from which the coeffi- 
cients a, [3, and y can be determined by the help of the Method 
of Least Squares. For ordinary ranges of velocity y is usually 
a small quantity, and it is often taken as zero. 

A current meter cannot be used for determining the ve- 
locity in a small trough, for the introduction of it into the 
■cross-section would contract the area and cause a change in the 
velocity in front of the wheel. In large conduits, canals, and 
rivers it is, however, one of the most convenient and accurate 
instruments. By holding it at a fixed position below the sur- 
face the velocity at that point is found ; by causing it to de- 
scend at a uniform rate from surface to bottom the mean ve- 
locity in that vertical is obtained ; and by passing it at a uni- 
form rate over all parts of the cross-section of a channel the 
mean velocity v is directly determined. It is usually mounted 
at the end of a long pole, which is graduated so that the depth 
of the meter below the water surface can be directly read.* 

Prob. 135. By rating a certain water meter, the equation 
V= 0.159+ 1905;/ was deduced for velocities varying from 
1 to 7 feet per second. Compute the velocity of the current 
^hen the wheel revolves 101 times in 41 seconds. 

*See paper by Stearns in Transactions American Society of Civil En- 
gineers, 1883, vol. xii. p. 301, for detailed account of the use of the current 
meter in the Sudbury conduit. 



256 FLOW IN XI VERS. [Chap. IX. 

Article iio. Floats. 

The method for measuring the discharge of streams which 
has been most extensively used is by observing the velocity of 
flow by the help of floats. Of these there are three kinds, sur- 
face floats, double floats, and rod floats. Surface floats should 
be sufficiently submerged so as to thoroughly partake of the 
motion of the upper filaments, and should be made of such a 
form as not to be readily affected by the wind. The time of 
their passage over a given distance is determined by two ob- 
servers at the ends of a base on shore by stop-watches ; or only 
one watch may be used, the instant of passing each section 
being signalled to the time-keeper. If / be the length of the 
base, and t the time of passage in seconds, the velocity of the 

float per second is 

/ 
v = — . 
t 

The numerical work of division can here, as in other cases, be 
best performed by taking the reciprocal of / from a table, and 
multiplying it by /, which for convenience may be an even 
number, such as 100 or 200 feet. 

A sub-surface float consists of a small surface float con- 
nected by a fine cord or wire with the large real float which is 
weighted so as to remain submerged, and keep the cord rea- 
sonably taut. The surface float should be made of such a 
form as to offer but slight resistance to the motion, while the 
lower float is large, it being the object of the combination to 
determine the velocity of the lower one alone. This arrange- 
ment has been extensively used, but it is probable that in all 
cases the velocity of the large float is somewhat affected by 
that of the upper one, as well as by the friction of the cord. 
In general the use of these floats is not to be encouraged, if 
any other method of measurement can be devised. 



Art. i io.] FLOATS. 257 

The rod float is a hollow cylinder of tin, which can be 
weighted by dropping in pebbles or shot so as to stand vertical- 
ly at any depth. When used for velocity determinations they 
are weighted so as to reach nearly to the bottom of the chan- 
nel, and the time of passage over a known distance determined 
as above explained. It is often stated that the velocity of a 
rod float is the mean velocity of all the filaments in the verti- 
cal plane in which it moves. Theoretically this is not the case ; 
and experiments by FRANCIS have proved that the velocity of 
the rod is usually from 1 to 5 per cent less than that of the 
mean velocity in the vertical. Francis has also deduced the 
following empirical formula for finding the mean velocity V nt 
from the observed velocity V r of the rod, 



V. 



= v r ( 1.012- 




in which d is the total depth of the stream, and d' the depth of 
water below the bottom of the rod.* This expression is prob- 
ably not a valid one, unless d' is less than about one-quarter of 
d\ usually it will be best to have d' as small as the character of 
the bed of the channel will allow. 

The log used by seamen for ascertaining the speed of ves- 
sels may be often conveniently used as a surface float when 
rough determinations only are desired, it being thrown from a 
boat or bridge. The cord of course must be previously 
stretched when wet, so that its length may not be altered by 
the immersion ; if graduated by tags or knots in divisions of 
six feet, the log may be allowed to float for one minute, and 
then the number of divisions run out in this time will be ten 
times the velocity in feet per second. 

The determination of particular velocities in streams by 
means of floats appears to be simple, but in practice many 

* Lowell Hydraulic Experiments, 4th Edition, p. 195. 



2;8 



FLOW IN RIVERS. 



[Chap. IX. 



uncertainties are found to arise, owing to wind, eddies, local 
currents, etc., so that a number of observations are generally 
required to obtain a precise mean result. For conduits, canals, 
and for many rivers the use of a current meter will be found 
to be more satisfactory and less expensive if many observa- 
tions are required. 

Prob. 136. A rod float runs a distance of 100 feet in 42 sec- 
onds, the depth of the stream being 6 feet, while the foot of 
the rod is 6 inches above the bottom. Compute the mean 
velocity in the vertical. 




Fig. 



Article hi. Other Current Indicators. 

Pitot's tube is an instrument for measuring the velocity of 
a current by the velocity-head which it will produce. In its 

simplest form it consists of a bent 
glass tube as shown in Fig. 74, in 
which the mouth of the submerged 
part is placed so as to directly face 
the current. The water then rises 
in the vertical part to a distance 1i 
above the surface of the flowing stream, and the velocity is ap- 
proximately equal to \ 2gh. The only advantage of this in- 
strument is that no time observation is necessary ; the disad- 
vantages are many, the chief being that the distance // is always 
very small, so that errors are liable to be made in determining 
its value. As actually constructed, PlTOT's apparatus generally 
consists of two tubes placed side by side with their submerged 
mouths at right angles, so that when one is opposed to the cur- 
rent, as seen in Fig. 74, the other stands normal to it, and the 
water surface in the latter tube hence is at the same level as 
that of the stream. Both tubes are provided with cocks which 
may be closed while the instrument is immersed, and it can be 
then lifted from the water and the head h be read at leisure. 



Art. hi.] OTHER CURRENT INDICATORS. 259 



It is found that the actual velocity is always less than \2gh i 
and that a coefficient must be deduced for each instrument by 
moving it in still water at known velocities. PlTOT's tube has 
been but little used, and is generally regarded as an imperfect 
instrument for velocity determinations. 

The hydrometric pendulum, shown also in Fig. 74, consists 
of a ball suspended from a string, which by the pressure of the 
current is kept at a certain inclination from the vertical, the 
angle of inclination being read on a graduated arc. The rela- 
tion between this angle and the velocity of the current must 
be determined experimentally before the instrument can be used 
in actual observations. This apparatus was employed by some 
of the early experimenters, but has now gone out of use. 

The hydrometric balance is similar in principle to the pen- 
dulum, the string being replaced by a rigid rod which is con- 
nected with a lever at its upper end, upon which weights are 
hung so as to keep the rod in a vertical position. The weights 
measure the intensity of the pressure of the current, and hence 
its velocity, the relation between them being first experimentally 
established for each instrument. The hydrometric balance is a 
mere curiosity, and has never been practically used for velocity 
determinations. A torsion balance, in which the pressure of 
the current on a submerged plate causes a spring to be tight- 
ened, has also been devised. All the instruments mentioned 
in this article are adapted only to the measurement of veloci- 
ties in small troughs or channels, and even for these have been 
but little used. 

Prob. 137. If the head h in a PlTOT tube is 0.0 1 feet, what 
is the approximate velocity of the current? If an error of 25 
per cent be made in reading h, how does this affect the deduced 
value of the velocity? 



26o 



FLOW IN RIVERS. 



[Chap. IX. 




Fig. 75. 



Article 112. Gauging the Flow. 

The most common method of gauging the flow of a stream 
which is too large to be measured by a weir will now be ex- 
plained. It involves field operations which, although simple 

in statement, generally re- 
quire considerable care and 
expense. In all cases the 
first step should be to estab- 
lish a water gauge whose 
zero is located with reference to a permanent bench mark, so 
that the stage of water at any time may be determined. Such 
a gauge is usually graduated to tenths of feet, intermediate 
values being estimated to hundredths. 

One or more sections a*t right angles to the direction of the 
current are to be established, and soundings taken at intervals 
across the stream upon them, the water gauge being read while 
this is done. The distances between the places of sounding 
are measured either upon a cord stretched across the stream, or 
by other methods known to surveyors. The data are thus ob- 
tained for obtaining the areas a 1 , a 2 , a 3 , etc., shown upon Fig. 
75, and the sum of these is the total area a. Levels should be 
run out upon the bank beyond the water's edge, so that in case 
of a rise of the stream the additional areas can be deduced. 
If a current meter is used, but one section is needed ; if floats 
are used, at least two are required, and these must be located at 
a place where the channel is of as uniform size as possible. 

The mean velocities i\ , z/ a , v 3 , etc., in each of the sections 
are next to be determined for each of the sub areas. If a cur- 
rent meter is used, this may be done by starting at one side of a 
subdivision, and lowering it at a uniform rate until the bottom 
is nearly reached, then moving it a foot or two horizontally and 
raising it to the surface, and continuing until the area has been 
covered. The velocity then deduced from the whole number 



Art. ii2.] GAUGING THE FLOW. 26 1 

of revolutions is the mean velocity for the subdivision. Or the 
meter may be simply raised and lowered in a vertical at the 
middle of the sub-area, and the result will be a close approxi- 
mation to the mean velocity. If rod floats are used they are 
started above the upper section, and the times of passing to the 
lower one noted, as explained in Art. no, the velocity deduced 
from a float at the middle of a sub-area being taken as the 
mean for that area. It will be found that the rod floats are 
more or less affected by wind, whose direction and intensity 
should hence always be noted. 

The discharge of the stream is the sum of the discharges 
through the several sub-areas, or 

q = a x v x + a^ -f a z v z + etc. ; 

and if this be divided by the total area a, the mean velocity 
for the entire section is determined. 

The following notes give the details of a gauging of the 
Lehigh River, near Bethlehem, Pa., made Oct. 15, 1885, in the 
above manner by the use of rod floats.* The two sections 
were 100 feet apart, divided into 10 equal divisions, each 30 feet 
in width. In the second column are given the soundings in 
feet taken at the upper section, in the third the mean of 



Subdivisions. 


Depths. 


Areas. 


Times, 


Velocities. 


Discharges. 


I 


O.O 


55-5 


380 


O.263 


14.6 


2 


3-0 

6.0 


I48.5 


220 


454 


67.4 


3 


201.7 


185 


0.540 


10S.9 


4 
5 


7-i 

7.o 
7.o 


217.5 
2IO.O 


I20 

145 


0.833 
0.690 


l8l. 2 
I44.9 


6 


186.O 


I50 


0.667 


I24. I 


7 


5-3 
4-3 
3.o 
2.2 
0.0 

a - 


150.8 


165 


0.606 


9I.4 


8 


114. 


200 


0.500 


57-o 


9 
10 


84.O 
42.O 


320 
430 


0.313 
0.233 


26.3 
9.8 




= 1410.O 


7= 825.6 



Journal of Engineering Society of Lehigh University, 1S85, vol. i. p. 75. 



262 FLOW IN RIVERS. [Chap. IX. 

the two areas in square feet, in the fourth the times of passage 
of the floats in seconds, in the fifth the velocities in feet per 
second, which are directly deduced from the times without ap- 
plying the correction indicated in Art. no, and in the last are 
the products a 1 v 1 , a^v^ , which are the discharges for the sub- 
divisions a iy <2 2 , etc. The total discharge is found to be 826 
cubic feet per second, and the mean velocity is 

826 

v == = 0.59 feet per second. 

1410 Dy r 

A second gauging of the stream, made a week later, when the 
water level was 0.59 feet higher, gave for the discharge 1336 
cubic feet per second, for the total area 1630 square feet, and 
for the mean velocity 0.82 feet per second. These results for 
discharge and velocity should probably be increased about 3 
per cent, in order to allow for the difference between the ve- 
locities as observed by the rod floats, and the true mean veloci- 
ties in the middle of the sub-areas. 

As to the accuracy of the above method, it may be said 
that with ordinary work, using rod floats, the discrepancies in 
results obtained under different conditions ought not to exceed 
10 per cent ; and with careful work, using current meters, they 
may often be of a much higher degree of precision. In any 
event the results derived from such gaugings of rivers are more 
reliable than can be obtained by any other method. 

Prob. 138. Compute the mean depth and the hydraulic 
radius for the above section of the Lehigh River. 



'^- 



Article 113. Gauging by Surface Velocities. 

If by any means the mean velocity v of a stream can be 
found, the discharge is known from the relation q = az\ the 
area a being measured as explained in the last article. An ap- 
proximate value of v may be ascertained by one or more float 



Art. 113. ] GAUGING BY SURFACE VELOCITIES. 263 

measurements by means of the known relations between it and 
the surface velocities. 

The ratio of the mean velocity v to the maximum surface 
velocity Fhas been found to usually lie between 0.7 and O.85, 
and about 0.8 appears to be a rough mean value. Accordingly, 

v = o.SV\ 

from which, if V be accurately determined, v can be computed 
with an uncertainty usually less than 20 per cent. 

Many attempts have been made to deduce a more reliable 
relation between v and V. The following rule derived from 
the investigations of BAZIN makes the relation dependent on 
the coefficient c, whose value for the particular stream is to be 
obtained from the evidence presented in the last chapter : 

V 

v = . 

■+? 

It is probable, however, that the relation depends more on the 
hydraulic radius and the shape of the section than upon the 
degree of roughness of the channel, which c mainly represents. 

The ratio of the mean velocity v x in any vertical to its sur- 
face velocity V 1 is less variable, lying between 0.85 and 0.92, so 

that 

v x = o. 9 V x 

may be used with but an uncertainty of a few per cent. If 
several velocities V lf V 2 , etc., be determined by surface floats, 
the mean velocities v x , v 2 , etc., for the several sub-areas a x , # 2 , 
etc., are known, and the discharge is q — a x v x -f- a,i\ -|- etc., 
as before explained. This method will usually prove unsatis- 
factory as compared with the use of rod floats. 

Since the maximum surface velocity is greater than the 
mean velocity v, and since the velocities at the shores are 



264 FLOW IN RIVERS. [Chap. IX. 

usually very small, it follows that there are in the surface two 
points at which the velocity is equal to v. If by any means 
the location of either of these could be discovered, a single 
velocity observation would give directly the value of v. The 
position of these points is subject to so much variation in 
channels of different forms, that no satisfactory method of lo- 
cating them has yet been devised. 

The influence of wind upon the surface velocities is so great, 
that these methods of determining v will prove useless, except 
in calm weather. A wind blowing up stream decreases the sur- 
face velocities, and one blowing down stream increases them, 
without materially affecting the mean velocity and discharge. 

Prob. 139. A stream 60 feet wide is divided into three sec- 
tions, having the areas 32, 65, and 38 square feet, and the surface 
velocities near the middle of these are found to be 1.3, 2.6, and 
1.4 per second. What is the approximate mean velocity of the 
stream ? 

Article 114. Gauging by Sub-surface Velocities. 

By means of a sub-surface float, or by a current meter, the 
velocity V at mid-depth in any vertical may be measured. 
The mean velocity v 1 in that vertical is very closely 

v, = 0.98 V. 

In this manner the mean velocities in several verticals across 
the stream may be determined by a single observation at each 
point, and these may be used, as in Art. 112, in connection 
with the corresponding areas to compute the discharge. 

It was shown by the observations of Humphreys and Abbot 
on the Mississippi that the velocity V is practically unaffected 
by wind, the vertical velocity curves for different intensities of 
wind intersecting each other at mid-depth. The mid-depth 
velocity is therefore a reliable quantity to determine and use, 



Art. 114.] GAUGING BY SUB-SURFACE VELOCITIES. 265 

particularly as the corresponding mean velocity v 1 for the 
vertical rarely varies more than 1 or 2 per cent from the 
value 0.98 V . 

The following relations between velocities in the cross- 
section were also deduced by HUMPHREYS and ABBOT.* The 
curve of velocities in any vertical was found to be a parabola 
whose mean equation is 



V — 3.26 — 0.7922 v 

in which V is the velocity at any distance y above or below the 
horizontal axis of the parabola, and d is the depth of the water 
at the point considered ; the axis being at the distance o.2gyd 
below the surface. The depth of the axis was found to vary 
greatly with the wind, an up-stream wind of force 4 depressing 
it to mid-depth, and a down-stream wind of force 5.3 elevating 
it to the surface. The velocity Fat any depth d f was shown 
to be related to the maximum velocity V m in that vertical by 
the equation 



wfi-4 



in which v is the mean velocity for the entire cross-section, and 

1.69 

These relations and many others which were deduced are very 
interesting, but are of little value in the actual gauging of 
streams. 

Prob. 140. Show that the vertical velocity formula of HUM- 
PHREYS and ABBOT can be put in the form 

V = 3. 19 + 0.47 1 - d -0.792(3, 

in which x is the depth below the surface. 

* Physics and Hydraulics of the Mississippi River, 2d Edition, 1876. 



266 FLOW IN RIVERS. [Chap. IX. 

Article 115. Comparison of Methods. 

This chapter, together with those preceding, furnishes many 
methods by which the quantity of water flowing through an 
orifice, pipe or channel, may be determined. A few remarks 
may now properly be made by way of summary. 

The method of direct measurement in a tank is always 
the most accurate, but except for small quantities is expensive, 
and for large quantities is impracticable. Xext in reliability 
and convenience come the methods of gauging by orifices and 
weirs. An orifice one foot square under a head of 25 feet will 
discharge about 40 cubic feet per second, which is as large a 
quantity as can be usually profitably passed through a single 
opening. A weir 20 feet long with a depth of 2.0 feet will 
discharge about 200 cubic feet per second, which may be taken 
as the maximum quantity that can be conveniently thus 
gauged. The number of weirs may be indeed multiplied for 
larger discharges, but this is usually forbidden by the expense 
of construction. Hence for larger quantities of water indirect 
methods of measurement must be adopted. 

The formulas deduced for the flow in pipes and channels 
in Chaps. VII and VIII enable an approximate estimation of 
their discharge to be determined when the coefficients and 
data which they contain can be closely determined. The re- 
marks in Art. 106 indicate the difficulty of ascertaining these 
data for streams, and show that the value of the formulas lies 
in their use in cases of investigation and design rather than for 
precise gaugings. For small pipes an accurately rated water 
meter is a cheap and convenient method of measuring the dis- 
charge, while for large pipes it will often be found difficult to 
devise an accurate and economical plan for precise determina- 
tions, unless the conditions are such that the discharge may be 
made to pass over a weir or be retained in a large reservoir 



Art. 115.] COMPARISON OF METHODS. 267 

whose capacity is known for every tenth of a foot in depth. 
For large aqueducts and for canals and streams the only 
available methods are those explained in this chapter. 

Surface floats are not to be recommended except for rude 
determinations, because they are affected by wind, and because 
the deduction of mean velocities from them is always subject 
to much uncertainty. Nevertheless many cases arise in prac- 
tice where the results found by the use of surface floats are 
sufficiently precise to give valuable information concerning 
the flow of streams. 

The double float for sub-surface velocities is used in deep 
and rapid rivers, where a current meter cannot be well operated 
on account of the difficulty of anchoring a boat. In addition to 
its disadvantages already mentioned may be noted that of ex- 
pense, which becomes large when many observations are to 
be taken. 

The method of determining the mean velocities in vertical 
planes by rod floats is very convenient in canals and channels 
which are not too deep or too shallow. The precision of a 
velocity determination by a rod float is always much greater 
than that of one taken by the double float, so that the former 
is to be preferred when circumstances will allow. 

Current-meter observations are those which now take the 
highest rank for precision and rapidity of execution. The first 
cost of the outfit is greater than that required for rod floats, 
but if much work is to be done it will prove the cheapest. 
The main objection is to the errors which may be introduced 
from the lack of proper rating : this is required to be done at 
regular intervals, as it is found that the relation between the 
velocity and the recorded number of revolutions sometimes 
changes during use. 



268 FLOW IN RIVERS. [Chap. IX. 

In the execution of hydraulic operations which involve the 
measurement of water a method is to be selected which will 
give the highest degree of precision with a given expenditure, 
or which will secure a given degree of precision at a minimum 
expense. Any one can build a road, or a water supply-system ; 
but the art of engineering teaches how to build it well, and at 
the least cost of construction and maintenance. So the science 
of hydraulics teaches the laws of flow and records the results 
of experiments, so that when the discharge of a conduit is to 
be measured or a stream is to be gauged the engineer may 
select that method which will furnish the required information 
in the most satisfactory manner and at the least expense. 

Prob. 141. Devise a method for measuring the velocity of 
a current different from any described in the preceding pages. 

Article 116. Variations in Velocity and Discharge. 

When the stage of water rises and falls a corresponding in- 
crease or decrease occurs in the velocity and discharge. The 
relation of these variations to the change in depth may be 
approximately ascertained in the following manner, the slope 
of the water surface being regarded as remaining uniform : 
Let the stream be wide, so that its hydraulic radius is nearly 
equal to the mean depth d\ then 

v = c yds = cs^d*. 
Differentiating this with respect to v and d gives 

dv = wes^d'^Sd = \v — — , 
d 

or 

6v 1 Sd 



v 



d 



Here the first member is the relative change in velocity when 
the depth varies from d to d ± Sd, and the equation hence 



Art. ii6.] VARIATIONS IN VELOCITY AND DISCHARGE. 269 

shows that the relative change in velocity is one-half the rela- 
tive change in depth. For example, a stream 3 feet deep, and 
with a mean velocity of 2 feet per second, rises so that the 
depth is 3.3 feet ; then 

dv = 2 X - X — = 0.1, 
2 3 

and the velocity becomes 2 + 0.1 =2.1 feet per second. This 
conclusion is of course the more accurate the smaller the varia- 
tion 6d. 

In the same manner the variation in discharge may be 
found. Thus : let b be the breadth of the stream, then 

q — cbdVds = cb$d*\ 

Sq = %cbsU^Sd\ 

dq __ 3 6d 
q 2d' 

Hence the relative change in discharge is \\ times that of the 
relative change in depth. This rule, like the preceding, sup- 
poses that 6d is very small, and will not apply to large varia- 
tions. 

The above conclusions may be expressed as follows : If the 
mean depth changes 1 per cent, the velocity changes 0.5 per 
cent, and the discharge changes 1.5 per cent. They are only 
true for streams with such cross-sections that the hydraulic 
radius may be regarded as proportional to the depth, and even 
for such sections are only exact for small variations in d and v. 
They also assume that the slope s remains the same after the 
rise or fall as before ; this will be the case if a condition of 
permanency is established, but, as a rule, while the stage of 
water is rising the slope is increasing, and while falling it is 
decreasing. 



270 FLO W IN RIVERS. [Chap. IX. 

Prob. 142. A stream of 4 feet mean depth delivers 800 
cubic feet per second. What will be the discharge when the 
depth is decreased to 3.9 feet? 



Article 117. Non-uniform Flow. 

In all the cases thus far considered, the slope of the channel, 
its cross-section, and the depth of the water have been regard- 
ed as constant. If these are variable along different reaches 
of the channel the case is one of non-uniformity, and the pre- 
ceding discussions do not apply except to the single reaches. 
The flow being permanent, the same quantity of water passes 
each section per second, but its velocity and depth vary as 
the slope and cross-section change. To discuss this case let 
there be several lengths, l 17 / 2 , . . . , l n , which have the falls 
k r , k 23 . . . , /i n , the water sections being a 1} a n _, . . . , a n , 
the wetted perimeter p l , p 2 , . . . , p, n and the velocities i\ , v n _ , 
. . . , v n . The total fall h x -j- h 2 -f- . . . -J- h n is expressed by 

h. Now the head corresponding to the mean velocity in the 

2 
first section is — . The theoretic head for the last section is 

V 2 "f-'n 

h A — . while the actual velocity-head is — . The difference 

2g ?g 

between these is the head lost in friction, or 



■^ Ocr O cr n "? or ' /Z o <r ' n n "* cr 

~& ~<b "1 -<b "2 ~e> Ll n -& 

in which f x , f t , . . . , f n are the friction factors for the differ- 
ent sections and surfaces, whose values in terms of the velocity 
coefficient c are, as seen from Art. 94, 



2P* lor oo- 

f — z&- f — ?&- f — r^_ 



Art. 117.] NON- UNIFORM FLOW. 2JI 

Let q be the discharge per second ; then, as the flow is perma- 
nent, 

q q q 

*'=-s:.' v > = ^' ■■■' *"=-»■ 

Inserting in the equation these values of /and v, it becomes 

2g \ aj a* J ' y \c*a? ' c?a* ' ' c„aj) 

which is a fundamental formula for the discussion of the flow 
in non-uniform channels. Since the values of c given in this 
chapter are for English feet, the data of numerical problems 
•can be inserted only when expressed in the same unit. 

The above discussion shows that the discharge q is a con- 
sequence, not only of the total fall h in the entire length of 
the channel, but also of the dimensions of the various cross- 
sections. The assumption has been made that a and / are 
constant in each of the parts considered ; this can be realized 
by taking the lengths / a , / 2 , etc., sufficiently short. If only 
one part be considered in which a and / are constant, a n — a x , 
all the terms but one in the second member disappear, and the 
two equations reduce to the simple formulas previously de- 
duced for the velocity and discharge in a uniform channel. 

An interesting problem is that where the flow is non- 
uniform in a channel of constant slope and section, which may 
be caused by an obstruction in the stream above or below the 
part considered. Here let a x and a. 2 be two sections whose 
distance apart is /, and let z\ and z> 2 be the mean velocities in 
those sections. Then if a and p be average values of the 
wetted area and perimeter, the formula becomes 

2g\a? a? ] 



272 



FLOW IN RIVERS. 



[Chap. IX.. 




Fig. 76. 



from which q can be computed when the other quantities are 

known. The important problem, 
however, is to discuss the change 
in depth between the two sec- 
tions. For this purpose let 
A X A 2 in Fig. 76 be the longi- 
tudinal profile of the water sur- 
face, let A^D be horizontal, and A Y C be drawn parallel to the 
bed BJB^. The depths A l B x ax\d A^B^ are represented by d x 
and d^ , the latter being taken as the larger. Let i be the con- 
stant slope of the bed B X B % ; then DC = il, and since DA 2 =± h 
and A 7 C == d 3 — d lf 

h = il — (d 2 — d x ). 

Inserting this value of h in the above equation and solving for 
/, there results 



1 = 



2P-W a? I 



g 
c 2 a 3 



(76) 



from which the length / corresponding to a change in depth 
d 2 — d l can be approximately computed. This formula is the 
more accurate the shorter the length /, since then the mean 
quantities/ and a can be obtained with greater precision, and 
c is subject to less variation. 



The inverse problem, to find the change in depth when / is 
given, cannot be directly solved by this formula, because the 
areas are functions of the depths. If the change is not great, 
however, a solution may be effected for the case of a channel 
whose breadth b is constant by regarding/ and a as equal to p x 
and a 1 ; and also by putting 

j_ 2_ a *- a * - 4 2 - d * - (4+4X^-4) _ 2 (4-4) 

* ' ~ Z 7 ~ a;a: ~ b a d,*d 9 * ~ b'd* ~ b*d? " 



Art. 118.] THE SURFACE CURVE. 273 

The formula then becomes 



d.-d 



I 

i — 



2 3 

'A- (76)' 



from which d 2 can be approximately computed when all the 
other quantities are given. 

Fig. 76 is drawn for the case of depth increasing down 
stream, but the reasoning is general, and the formulas apply 
equally well when the depth decreases. In the latter case the 
point A 2 is below C, and d 2 — d 1 will be found to be nega- 
tive. As an example, let it be required to determine the de- 
crease in depth in a rectangular conduit 5 feet wide and 333 
feet long, which is laid with its bottom level, the depth of water 
at the entrance being maintained at 2 feet, and the quan- 
tity supplied being 20 cubic feet per second. Here / = 333, 
b = 5, d x = 2, p x = 5 -|- 4 = 9, q = 20, and i = o. Taking 
c = 89, and substituting all values in the formula, there is 
found d 2 — d 1 — — 0.16 feet, whence d 2 = 1.84 feet, which is 
to be regarded as an approximate probable value. It is likely 
that values of d 2 — d x computed in this manner are liable to an 
uncertainty of 10 or 20 per cent, the longer the distance / the 
greater being the error of the formula. In strictness also c 
varies with depth, but errors from this cause are small when 
compared to those arising in selecting its value. 

Prob. 143. Compute the value of d 2 for the above example 
when the bed of the conduit has the uniform slope i = 0.01. 

Ans. d 2 — 5.38 feet. 

Article 118. The Surface Curve. 

In the case of uniform flow the slope of the water surface is 
parallel to that of the bed of the channel, and the longitudinal 



274 



FLOW IN FIVERS. 



[Chap. IX. 



profile of the water surface is a straight line. In non-uniform 
flow, however, the slope of the water surface continually varies, 
and the longitudinal profile is a curve whose nature is now to 
be investigated. As in the last article, the slope i of the bed 
of the channel will be taken as constant, and its cross-section 
will be regarded as rectangular. Moreover, it will be assumed 
that the stream is wide compared to its depth, so that the 
wetted perimeter may be taken as equal to the width and the 
hydraulic radius equal to the mean depth (Art. 93). These 
assumptions are closely fulfilled in many canals and rivers. 

The last formula of the preceding article is rigidly exact if 

the sections a x and <? 2 are consecutive, so that / becomes dl 

and d 2 — d x becomes Sd. Making these changes, and placing 

P l 

— equal to -=, in accordance with the above assumptions, the 

formula becomes 



6d 
Si 



ctfd 



(77) 



zb 2 d- 



in which d is the depth of the water at the place considered. 
This is the general differential equation of the surface curve. 



To discuss this curve let D be the depth of the water if the 

flow were uniform. The slope 
s of the water surface would 
then be equal to the slope i of 
the bed of the channel and 
from the general equation for 
mean velocity. 



q — av — cbD \ "ri = cbD S Di. 

Inserting this value of q the 
equation reduces to 




Art. 118.] THE SURFACE CURVE. 2?$ 

61 ~ l c 2 i £> 3 ' {77) 

I — 71 

g d 

in which d and / are the only variables, the former being the 
ordinate and the latter the abscissa, measured parallel to the 
bed BB, of any point of the surface curve. 

First, suppose that D is less than d, as in the upper diagram 
of Fig. 77, where AA is the surface curve under the non-uni- 
form flow, and CC is the line which the surface would take in 
case of uniform flow. The numerator of (77)' is then positive, 
and the denominator is also positive, since i is very small. 
Hence 6d is positive, and it increases with d in the direction 
of the flow ; going up stream it decreases with d lf and the sur- 
face curve becomes tangent to CC when d = D. This form of 
curve is that usually produced above a dam, and is called the 
curve of backwater. 

Second, let d be less than D, as in the second diagram of 
Fig. 74. The numerator is then negative, and the denomi- 
nator positive ; dd is accordingly negative, and A A is concave 
to the bed BB, whereas in the former case it was convex. 
This form of surface curve may occur when a sudden fall exists 
in the stream below the point considered ; it is of slight practi- 
cal importance compared to the previous case. 

A very curious phenomenon is that of the so-called "jump" 
which sometimes occurs in shallow channels, as shown in Fig. 
78. This happens when the de- 
nominator in {yyy is zero, the 3^7 
numerator being positive ; then 
dd 



becomes infinite, and the 

SI Fig. 78. 




276 FLOW IN RIVERS. [Chap. IX. 

water surface stands normal to the bed. Placing the denomi- 
nator of (77) equal to zero, there is found 

q 1 — gb*d z or v 2 — gd. 

Now by further consideration it will appear that the varying 
denominator in passing through zero changes its sign. Above 
the jump where the depth is d l the velocity is greater than 
Vgd 1 , and below it is less than Vgd„ . The condition for the 
occurrence of the jump is that an obstruction should exist in 
the stream below, that the slope i should not be small, and 
that the velocity should be greater than \ r gd 1 . To find the 
slope i which is necessary, 

v 1 =cVd 1 i v*>gd^ whence i >^j* 

Hence the jump cannot occur when i is less than ~. For an 

unplaned plank trough c may be taken at about 100 ; hence the 
slope for this must be equal to or greater than 0.00322. 

To determine the height of the jump, or the value of d^ in 

terms of d. , it is to be observed that the lost head is — — , 

and that this is lost in two ways, first by the impact due to the 
enlargement of section (Art. 68), and second by the rising of the 
whole quantity of water through the height ^(d 2 — d^, the loss 
in friction in the short distance between d l and d 2 being neg- 
lected. Hence 

v_l - v? = (i\ - t' s ) a d,-d x 



Inserting in this the value of T' 2 , found from the relation 
v 7 d 2 — v x d x , dividing by d n — d lf and solving for d^ , gives 



d = 2A/d^ (78) 



Art. ng.] BACKWATER. 2J7 

The following is a comparison between the values of d 2 com- 
puted by this formula and the observed values in four experi- 
ments made by BlDONE, the depths being in feet : 



dl. 


Vi . 


Observed d* . 


Computed d 2 


0.149 


4.59 


O.423 


0-439 


0.154 


4-47 


O.421 


0.437 


0.208 


5-59 


O.613 


O.636 


0.246 


6.28 


0739 


0.777 



The agreement is very fair, the computed values being all 
slightly greater than the observed, which should be the case, 
because the above reasoning omits the frictional resistances 
between the points where d x and d^ are measured. 

Prob. 144 Discuss formula (78) by placing for q its value 
cbd Vds, where s is the slope of the water surface. 



Article 119. Backwater. 

When a dam is built across a channel the water surface is 
raised for a long distance up stream. This is a fruitful source 
of contention, and accordingly many attempts have been made 
to discuss it theoretically, in order to be able to compute the 
probable increase in depth at various distances back from a 
proposed darn. None of these can be said to have been suc- 
cessful except for the simple case where the slope of the bed 
of the channel is constant, and its cross-section such that the 
width may be regarded as uniform and the hydraulic radius 
be taken as equal to the depth. These conditions are closely 
fulfilled for many streams, and an approximate solution may be 
made by the formula (77) of Art. 118. It is desirable, how- 
ever, to obtain an exact equation of the surface curve, so as to 
secure a more reliable method. 



2/8 FLOW IN RIVERS. [Chap. IX. 

For this purpose the differential equation (yj)' of the last 
article may be written in the form 







/ 




C'l- 


61 


M 


i 


I — 


g 


6d ~ 


1 d" 








\ 


IT 


— i 



in which / and d are the co-ordinates of any point of the curve. 

d 
Let jr be the independent variable x, so that d = Dx\ then 

D . /?/ fi\ 6x 

61— — dx — -A I — — -i , 

i ' 2 \ g l x — I 

the general integral of which is 



2 2 



^A/l , X*-\-X-\-I I 2^+l\ 

— — |( 2~ lo^ — - — - arc cot — I » 



which is the equation of the surface curve. To use this let 

— . z I<?S ^W or $ \~rjJ ^ e P ut a5 an a ^" 

u .<<^ breviation lor the logarithmic 

'^WM^M^Wm^^mm^^ and circular function in the 

FlG - 79- second member. Also let d 2 be 

the depth at the dam. and let / be measured up stream from 

that point to a section where the depth is d 1 . Then taking- 

the integral between these limits the equation becomes 

d, — d,, n /i r\ r . rd,\ d 9 \~\ 



'^+»&-i)\*.W-*® 



• 0-9) 



which is the practical formula for use. In like manner. d„ may 
represent a given depth at any section, and d 1 any depth farther 

up the stream. 

When d = D. or the depth of the backwater becomes equal 
to that of the previous uniform flow, x is unity, and hence / is 



Art. 119.] BACKWATER. 279 

infinity. The slope CC of uniform flow is therefore an asymp- 
tote to the backwater curve. Accordingly, no matter how 
little d t may exceed D, the depth d x is always greater than D, 
although it often happens for steep slopes that d x becomes 
practically equal to D at distances above d 2 , which are not 
great. In the investigations of backwater problems there are 
two cases: d 1 and d^ may be given and / is to be found, or / is 
given and one of the depths is to be found. To solve these 
problems, a table giving values of the backwater function 

<P[jy) will be found on the next page.* The argument of the 

table is -j , which being less than unity, is more convenient for 

d 
tabular purposes than yr , whose values range from o to 00. 

The following examples will illustrate the method of procedure. 

A stream of 5 feet depth is to be dammed so that the water 
just above the dam will be 10 feet. Its uniform slope is 
0.000189, or a little less than one foot per mile, and the surface 
of its channel is such that the coefficient c is 65. It is required 
to find the distance back from the dam at which the depth of 

water is 6 feet. Here d 2 — 10, d 1 = 6, D = 5, — = 0.5 for 

dA _ D 



which the table gives 01^) =0.1318, -j- = 0.833 f° r which 

the table gives (-^ j = 0.4792, and — "= 5291. These values 
inserted in the formula give 

65" 



1= (10 - 6)5291 + 5 ^5291 - -^-^J (0.4792 - 0.1318); 
/= 30 125 feet = 5.70 miles. 

*From Bresse's La Mecanique Appliquee (Paris, 1873), vol. ii. p. 556. 



280 FLOW IN RIVERS. [Chap. IX. 

TABLE XXII. VALUES OF THE BACKWATER FUNCTION. 



D 
d 


♦(£) 


D 

d 


Hi) 


D 
d 


Hi) 


D 

~d 


H£) 


I . 


00 O 


948 


0.8665 


O.815 


0.4454 


\ O.52 


o.i435 


O 


999 


2.1834 


946 


.8539 


.810 


•4367 


•51 


.1376 




993 


1-9523 


944 


.8418 


.805 


.4281 


•50 


• 1318 




997 


1. 8172 


942 


.8301 


.800 


.4198 


•49 


.1262 




996 


I.72T3 


940 


.8188 


•795 


.4117 


•43 


.1207 




995 


I.6469 


93S 


.8079 


.790 


• 4039 


• 47 


•ii54 




994 


I. 5861 


•936 


• 7973 


.7S5 


.3962 


.46 


.1102 




993 


1.5343 


•934 


.7871 


.7S0 


.38S6 


•45 


.1052 




992 


I .4902 


932 


• 7772 


•775 


.3313 


44 


.1003 




991 


I.45TO 


.930 


• 7675 


.770 


.3741 


•43 


•0995 




990 


I-4I59 


.92S 


• 7581 


•705 


• 3671 


•42 


.0909 




989 


I. 3841 


926 


.7490 


,760 


.3603 


.41 


.0S65 




988 


I- 3551 


924 


.7401 


•755 


• 3536 


•40 


.0821 




987 


1.3284 


922 


• 7315 


.750 


• 3470 


•39 


•0779 




986 


1.3037 


920 


.7231 


•745 


• 3406 


.36 


.075S 




9§5 


I.2807 


918 


• 7149 


.740 


•3343 


•37 


. 0699 : 




984 


I.2592 


916 


.7069 


• 735 


.32S2 


06 


.0660 




9 S 3 


I.239O 


914 


.6990 


•730 


.3221 


•35 


.0623 




9S2 


I .2199 


912 


.6914 


.725 


.3162 


•34 


•05S7 




gSr 


I. 2019 


910 


.6839 


.720 


•3104 


•33 


.0553 




980 


I.1848 


.908 


.6766 


.715 


•3047 


•32 


•0519 




979 


I.I6S6 


906 


.6695 


.710 


.2991 


•3i 


.04S6 




978 


I.I53I 


.904 


.6625 


•705 


•2937 


.30 


•0455 




977 


I-I333 


902 


.6556 


.70 


.2883 


•29 


.0425 




976 


I.I24I 


900 


.64S9 


.69 


•277S 


.28 


• 0395 




975 


I. I 105 


395 


.6327 


.63 


•2677 


.27 


.0367 




974 


I.O974 


S90 


•6173 


.67 


.25S0 


.26 


.0340 




973 


I.0S4S 


8S5 


.6025 


.66 


.2486 


.25 


.0314 




972 


I.0727 


83o 


•58S4 : 


.65 


•2395 


.24 


.0290 




97i 


I. o6lO 


S75 


•5749 


.64 


.2306 


.23 


.0266 




970 


I.0497 


S70 


.5619 


.63 


. 222 1 


.22 


•0243 




968 


1.0232 


S65 


• 5494 


.62 


.2138 


.21 


.0221 




966 


1.00S0 


860 


• 5374 


.61 


.205S 


.20 


.0201 




964 


0.9S90 


S55 


.525s 


.60 


. 19S0 


.18 


.OT62 


1 


962 


• 9709 


S50 


.5146 


•59 


.1905 


.16 


.0128 I 




960 


•9539 


S45 


• 5037 


-53 


.1832 


• 14 


.0098 




958 


• 9376 


S40 


.4932 


•57 


.1761 


.12 


.0072 


! 


956 


.9221 


S35 • 


.4831 


.56 


. 1692 


.10 


.0050 




954 


•9073 


S30 


• -I733 


•55 


.1625 


.06 


.001S 




952 


.8931 


S25 


.4637 


•54 


.1560 


.or 


.0001 | 




950 


•S795 


S20 


• 4544 


•53 


.1497 

1 


.00 


.0000 



Art. 119.] BACKWATER. 28 1 

In this case the water is raised one foot at a distance of 5.7 
miles up stream from the dam, in spite of the fact that the fall 
in the bed of the channel is nearly 5.7 feet. 

The inverse problem, to compute d 1 or d„ , when / and d^ or 

d x is given, can only be solved by repeated tentative trials by 

the help of Table XXII. For example, let / = 30 125 feet, the 

other data as above, and it be required to determine d % so that 

d 1 shall be only 5.2 feet, or 0.2 feet greater than the original 

D 5 
depth of 5 feet. Here -j = — = 0.962, and from the table 

0fyH = 0.9700. Then the formula becomes 



30 125 = (d, - 5.2)5291 + 5 X 5160 
which reduces to 



0.9700 - 0(2)-) J 



2 610 = 52914 — 25 8oo0U=J 



Values of d are now to be assumed until one is found which 



D 



satisfies this equation. Let d 2 = 8 feet, then -7- = 0.625, and 

fdA 
from the table (pijkj = 0.2179. Substituting these, 

32 610 == 42 328 — 5 622 = 36 706, 

which shows that the assumed value is too large. Again, take 

D (dX 

d 2 = 7 feet, then '-=- = 0.714, and from the table 0l-^J = 0.3036. 

whence 

32 610 = 37 037 — 7 833 = 29 204, 

which shows that 7 feet is too small. If d t = 7.4 feet, 

D . ■ . Jd. 

and then 



r = 0.675 and (pij. j = 0.2628, 



32 610 = 39 153 — 6 780 = 32 373, 



282 FLOW IN RIVERS. [Chap. IX. 

This indicates that 7.4 is a little too small, and on trying 7.5 it 
is found to be too large. The value of d, hence lies between 
7.4 and 7.5 feet, which is as close a solution as will generally be 
required. The height of the dam may now be computed by 
Art. 58, taking the rise d' at about 2.45 feet. 

In conclusion, it should be said that if the slope, width, or 
depth changes materially, the above method cannot be em- 
ployed in which the distance / is counted from the dam as an 
origin. In such cases the stream should be divided into reaches, 
for each of which these quantities can be regarded as constant. 
The formula can then be used for the first reach, and the depth 
at its upper section determined ; calling this depth d 2 , the ap- 
plication can then be made to the next reach, and so on in 
order. Strictly speaking, the coefficient c varies with the 
depth, and by Kutter's formula (Art. 10 1) its varying values 
may be ascertained, if it be thought worth the while. Even if 
this be done, the resulting computations must be regarded as 
liable to considerable uncertainty. In computing depths for 
given lengths probably an uncertainty of 10 per cent or more 
in values of d^ — d l should be expected. In regard to the 
depth D, it may be said that this should be determined by the 
actual measurement of the area and wetted perimeter of the 
cross-section during uniform flow, the hydraulic radius com- 
puted from these being taken as D. 

Prob. 145. A stream whose cross-section is 2400 square 
feet and wetted perimeter 300 feet has a uniform slope of 2.07 
feet per mile, and its condition is such that c = 70. It is pro- 
posed to build a dam which raises the water 6 feet above its 
former level, without increasing its width. Compute the 
amount of rise due to the backwater at distances of 1, 2, and 3 
miles up stream from the dam. 



Art. 120. j THEORETIC AND EFFECTIVE POWER. 283 



CHAPTER X. 
MEASUREMENT OF WATER POWER. 

Article 120. Theoretic and Effective Power. 

The theoretic energy of W pounds of water falling through 
h feet is Wh foot-pounds, and if this occurs in one second the 
energy per second is Wh, and the theoretic horse-power is 

Wh 

HP= = 0.001818^%. , . . . (80) 

550 . 

If this power could all be utilized it would be able to lift the 
same weight of water per second through the same vertical 
height, and an efficiency of unity would be secured. Owing 
to friction, impact, leakage, and other losses, the efficiency 
must always be less than unity. 

When the energy of a water-fall is to be transformed into 
useful work the water is made to pass over a wheel or through 
a hydraulic motor in such a manner that when the fall h has 
been accomplished the ,water has little or no velocity. If the 
water falls freely through the height h it acquires the velocity 
V2gh, and the energy is still potential, and equal to Wh. If, 
however, this velocity is destroyed, the energy is either trans- 
formed into heat or into useful work. In the case of flow 
through a long pipe nearly all the energy of the head h may 
be expended in heat in overcoming the frictional resistances; 
such a method of bringing water to a motor is therefore to 
be avoided. 

To utilize the energy of a water-fall in work the water is to 
be collected in a reservoir, canal, or head race, from which it is 



284 MEASUREMENT OF WATER POWER. [Chap. X. 

carried to the motor through a pipe, penstock, or flume, and 
after doing its work it issues into the tail race or lower level. 
In designing these constructions care should be taken to avoid 
losses in energy or head, and for this purpose the principles of 
the preceding chapters should be applied. The entrance from 
the head race into the penstock, and from the penstock to the 
motor, should be smooth and well rounded ; sudden changes in 
cross-section should be avoided, and all velocities should be low 
except that which is to be utilized in impulse. If these pre- 
cautions be carefully observed the loss in the head h outside of 
the motor can be made very small. 

The effective power of a water-fall, or that utilized by the 
motor, may be in the best constructions as high as 90 per cent 
of the theoretic power. In any case, if e be the efficiency and 
k the work actually obtained, 

k = eK = eWh. 

That hydraulic motor will be the best, other things being equal, 
which furnishes the highest value of e. In practice values of e 
are usually between the limits 0.25 and 0.90, the lower values 
occurring where a cheap and abundant water supply exists, so 
that sufficient power can be obtained with an inexpensive 
wheel, for it is a general rule that the cost of a hydraulic motor 
increases with the efficiency. 

There are to be distinguished two efficiencies — the effici- 
ency of the fall and the efficiency of the motor. The same 

expression 

k 

will apply to both, k being the effective work of the motor. 
The efficiency of the fall is that value of e found by using the 
actual weight W delivered per second, and the total height h 
from the water level in the head race to that in the tail race. 



Art. 121.] MEASUREMENT OF THE WATER. 285 

The efficiency of the motor is that value of e found by using 
the actual weight W which passes through the motor, and the 
effective head h that acts upon it. The second W may be less 
than the first on account of leakage, and the second h may be 
less than the first on account of losses of head. 

To determine the theoretic power in any case, it is only 
necessary to measure W and h and insert their values in for- 
mula (80). The two following articles will treat of these meas- 
urements, and the determination of the effective power of the 
motor will be discussed afterwards. The efficiency e is, then, 
the ratio of the effective power to the theoretic power, or the 
ratio of the effective work to the theoretic work. 

Prob. 146. A weir with end contractions and no velocity 
of approach has a length of 1.33 feet, and the depth on the 
crest is 0.406 feet. The same water passes through a small 
turbine under the effective head 10.49 ^ eet - Compute the 
theoretic horse-power. 

Article 121. Measurement of the Water. 

In order to determine the weight W which is delivered per 
second there must be known the discharge per second q, and 
the weight of a cubic unit of water, or 

W = wq. 

The quantity w is to be found by weighing very accurately one 
cubic foot, or any given volume of water, or by noting the tem- 
perature and using the table in Art. 3. In common approxi- 
mate computations w may be taken at 62.5 pounds per cubic 
foot. In precise tests of motors, however, its actual value 
should be ascertained as closely as possible. 

The measurement of the flow of water through orifices, 
weirs, tubes, pipes, and channels has been so fully discussed in 



286 MEASUREMENT OF WATER POWER. [Chap. X. 

the preceding chapters, that it only remains here to mention 
one or two simple methods applicable to small quantities, and 
to make a few remarks regarding the subject of leakage. In 
any particular case that method of determining q is to be se- 
lected which will furnish the required degree of precision with 
the least expense (Art. 115). 

For a small discharge the water may be allowed to fall into 
a tank of known capacity. The tank should be of uniform 
horizontal cross-section, whose area can be accurately deter- 
mined, and then the heights alone need be observed in order 
to find the volume. These in precise work will be read by 
hook gauges, and in cases of less accuracy by measurements 
with a graduated rod. At the beginning of the experiment a 
sufficient quantity of water must be in the tank so that a read- 
ing of the gauge can be taken ; the water is then allowed to 
flow in, the time between the beginning and end of the experi- 
ment being determined by a stop-watch, duly tested and rated. 
This time must not be short, in order that the slight errors in 
reading the watch may not affect the result. The gauge is 
read at the close of the test after the surface of the water be- 
comes quiet, and the difference of the gauge-readings gives the 
depth which has flowed in during the observed time. The 
depth multiplied by the area of the cross-section gives the vol- 
ume, and this divided by the number of seconds during which 
the flow occurred furnishes the discharge per second q. 

If the discharge be very small, it may be advisable to weigh 
the water rather than to measure the depths and cross-sections. 
The total weight divided by the time of flow then gives directly 
the weight IV. This has the advantage of requiring no tem- 
perature observation, and is probably the most accurate of all 
methods, but unfortunately it is not possible to weigh a con- 
siderable volume of water except at great expense. 



Art. I2i.] MEASUREMENT OF THE WATER. 287 

When water is furnished to a motor through a small pipe 
a water meter may often be advantageously used to determine 
the discharge. This consists of a box with two chambers, the 
water entering into one and passing out of the other. In going 
from the first to the second chamber the water moves a vane, 
a piston, a disk, or some other device, which communicates 
motion to a train of clockwork, and thereby causes pointers 
to move on dials. The external appearance of a water meter is 
similar to that of a gas meter, and it is read in the same way. 
No water meter, however, can be regarded as accurate until it 
has been tested by comparing the discharge as recorded by it 
with the actual discharge as determined by measurement or 
weighing in a tank. Such a test furnishes the constants for 
correcting the result found by its readings, which otherwise is 
liable to be 5 or 10 per cent in error. 

The leakage which occurs in the flume or penstock before 
the water reaches the wheel should not be included in the 
value of W, which is used in computing its efficiency. The 
manner of determining the amount of leakage will vary with 
the particular circumstances of the case in hand. If it be very 
small, it may be caught in pails and directly weighed. If large 
in quantity, the gates which admit water to the wheel may be 
closed, and the leakage being then led into the tail race it may 
be there measured by a w r eir, or by allowing it to collect in a 
tank. The leakage from a vertical penstock whose cross-sec- 
tion is known may be ascertained by filling it with water, the 
wheel being still, and then observing the fall of the water level 
at regular intervals of time. In designing constructions to 
bring water to a motor, it is best, of course, to arrange them so 
that all leakage will be avoided, but/ this cannot often be fully 
attained, except at great expense. 

The most common method of measuring q is by means of 
a weir placed in the tail race below the wheel. This has the 



2SS MEASUREMENT OF WATER POWER. [Chap. X.. 

disadvantage that it sometimes lessens the fall which would be 
otherwise available, and that often the velocity of approach is 
high. It has, however, the advantage of cheapness in construc- 
tion and operation, and for any considerable discharge appears 
to be almost the only method which is both economical and 
precise. 

Prob. 147. A vertical penstock whose cross-section is 15.98 
square feet is filled with water to a depth of 10.50 feet. Dur- 
ing the space of two minutes the water level sinks 0.02 feet. 
What is the leakage in cubic feet per second ? 



Article 122. Measurement of the Head. 

The total available head h between the surface of the water 
in the reservoir or head race and that in the lower pool or tail 
race is determined by running a line of levels from one to the 
other. Permanent bench marks being established, gauges can 
then be set in the head and tail races, and graduated so that 
their zero points will be at some datum below the tail-race 
level. During the test of a wheel each gauge is read by an 
observer at stated intervals, and the difference of the readings 
gives the head h. In some cases it is possible to have a float- 
ing gauge on the lower level, the graduated rod of which is 
placed alongside of a glass tube that communicates with the 
upper level ; the head h is then directly read by noting the 
point of the graduation which coincides with the water surface 
in the tube. This device requires but one observer, while the 
former requires two ; but it is usually not the cheapest arrange- 
ment unless a large number of observations are to be taken. 

When water is delivered through a nozzle or pipe to a hy- 
draulic motor the head which is to be determined for ascertain- 
ing the efficiency of the motor is not the total fall, since a large 
part of that may be lost in friction in the pipe, but is merely 



Art. 122.] MEASUREMENT OF THE HEAD. 289 



v 



the velocity-head — of the issuing jet. The value of v is 

known when the discharge q and the area of the cross-section 

of the stream have been determined, and 

2 2 

v q 



k = — = 



2g 2gd 2 



In the same manner when a stream flows in a channel against 
the vanes of an undershot wheel the effective head is the 
velocity-head, and the theoretic energy is 



K=Wk= W 



v wq 



2g 2ga l 



If, however, the water in passing through the wheel falls a 

distance /i below the mouth of the nozzle, then the effective 

head is 

/l = — + /*.. 
2 g 

In order to fully utilize the fall k Q it is plain that the wheel 
should be placed as near the level of the tail race as possible. 

When water enters upon a wheel through an orifice which is 
controlled by a gate, losses of head will result, which can be 
estimated by the rules of Chapters IV and V. If this orifice is 
in the head race the loss of head should be subtracted from the 
total head in order to obtain the h which really acts upon the 
wheel. But if the regulating gates are a part of the wheel 
itself, as is the case in a turbine, the loss of head should not be 
subtracted, because it is properly chargeable to the construc- 
tion of the wheel, and not to the arrangements which furnish 
the supply of water. In any event that head h should be 
determined which is to be used in the subsequent discussions : 
if the efficiency of the fall is desired, the total available head is 
required ; if the efficiency of the motor, that effective head is to 
be found which acts directly upon it (Art. 120). 



29O MEASUREMENT OF WATER POWER. [Chap. X. 

Prob. 148. A pressure gauge at the entrance of a nozzle 
registers 116 pounds per square inch, and the coefficient of 
velocity of the nozzle is 0.98. Compute the effective velocity- 
head of the issuing jet. 

Article 123. Measurement of Effective Power. 

The effective work and horse-power delivered by a water- 
wheel or hydraulic motor is often required to be measured. 
Water-power may be sold by means of the weight W, or quan- 
tity q, furnished under a certain head, leaving the consumer to 
provide his own motor ; or it may be sold directly by the num- 
ber of horse-power. In either case tests must be made from 
time to time in order to insure that the quantity contracted 
for is actually delivered and is not exceeded. It is also fre- 
quently required to measure effective work in order to ascertain 
the power and efficiency of the motor, either because the party 
who buys it has bargained for a certain power and efficiency, 
or because it is desirable to know exactly what the motor is 
doing in order to improve if possible its performance. 

The effective work of a motor might be measured if it could 
be used to operate a pump in which are no losses of any kind. 
This pump might raise the same water that drives the motor 
through a vertical height h x ; then the effective work per second 
would be Wh x , and the efficiency of the motor would be 

k Wh, h 



~ K~ Wh ~ h ' 
It is needless, however, to say that such a pump is purely im- 
aginary. 

A method in which the effective work of a small motor may 
be measured is to compel it to exert all its power in lifting a 
weight. For this purpose the weight may be attached to a 
cord which is fastened to the horizontal axis of the motor, and 
around which it winds as the shaft revolves. The wheel then 



Art. 123.] MEASUREMENT OF EFFECTIVE POWER. 29 1 

expends all its power in lifting this weight W 1 through the 
height Ji x in t 1 seconds, and the work performed per second 
then is 

* = — ■ 

This method, although practicable, is usually cumbersome in 
actual use, on account of the difficulty of determining t x with 
precision, since the height h x which can be secured is generally 
small. 

The usual way of measuring the effective power is by means 
of the friction brake or power dynamometer, which is described 
in the next article. By this method the effective work per second 
k is readily determined, and then the power of the motor is 

k 
hp — = 0.0018 1 8£, 

and its efficiency is found by dividing this by the theoretic 
power. 

The test of a hydraulic motor has for its object : first, the 
determination of the effective energy and power ; secondly, the 
determination of its efficiency; and third, the determination of 
the speed which gives the greatest power and efficiency. If 
the wheel be still, there is no power ; if it be revolving very fast, 
the water is flowing through it so as to change but little of its 
energy into work : and in all cases there is found a certain 
speed which gives the maximum power and efficiency. To 
execute these tests, it is not at all necessary to know how the 
motor is constructed or the principle of its action, although 
such knowledge is very valuable, and is in fact indispensable, 
in order to enable the engineer to suggest methods by which 
its operation may be improved. 

Prob. 149. What is the horse-power of a motor which in 
75.5 seconds lifts a weight of 320 pounds through a vertical 
height of 42 feet ? 



2C)^ 



MEASUREMEXT OF WATER POWER. 



[Chap. X. 



Article 124. The Friction Brake, or Power 
Dynamometer. 

The effective work k performed by a hydraulic motor is 
measured by an apparatus, invented by Prony, called the fric- 
tion brake. In Fig. 80 is illustrated a simple method of apply- 
ing it to a vertical shaft, the 
upper diagram being a plan and 
the lower an elevation. Upon 
the vertical shaft is a fixed 
pulley, and against this are seen 
two rectangular pieces of wood 
hollowed so as to fit it, and con- 
nected by two bolts. By turn- 
ing the nuts on these bolts while 
the pulley is revolving, the fric- 
tion can be increased at pleas- 
ure, even so as to stop the mo- 
tion ; around these bolts between 
the blocks are two spiral springs 
(not shown in the diagram) which press the blocks outward 
when the nuts are loosened. To one of these blocks is attached 
a cord which runs horizontally to a small movable pulley over 
which it passes, and supports a scale pan in which weights are 
placed. This cord runs in a direction opposite to the motion 
of the shaft, so that when the brake is tightened it is prevented 
from revolving by the tension caused by the weights. The 
direction of the cord in the horizontal plane must be such 
that the perpendicular let fall upon it from the centre of the 
shaft, or its lever arm. is constant ; this can be effected by 
keeping the small pointer on the brake at a fixed mark estab- 
lished for that purpose. 




Fig. 80. 



To measure the power of the wheel, the shaft is disconnected 
from the machinery which it usually runs, and allowed to 



Art. 124.] THE FRICTION BRAKE. 293 

revolve, transforming all its work into heat by the friction be- 
tween the revolving pulley and the brake which is kept station- 
ary by tightening the nuts, and at the same time placing 
sufficient weights in the scale pan to hold the pointer at the 
fixed mark. Let n be the number of revolutions per second as 
determined by a counter attached to the shaft, P the tension 
in the cord which is equal to the weight of the scale pan and 
its loads, / the lever arm of this tension with respect to the 
centre of the shaft, r the radius of the pulley, and F the total 
force of friction between the pulley and the brake. Now in 
one revolution the force F is overcome through the distance 
27tr, and in n revolutions through the distance 2nrn. Hence 
the effective work done by the wheel in one second is 

k = F . 2nrn = 27tn . Fr. 
The force F acting with the lever-arm r is exactly balanced by 
the force P acting with the lever-arm /; accordingly, 

Fr = Pl; 
and hence the effective work per second is 

k = 27cnPl, 

and the effective horse-power is 

27tnPl 
hp= = o.on^2nPl. .... (81) 

As the number of revolutions in one second cannot be accurately 
read, it is usual to record the counter readings every minute or 
half minute ; if N be the number of revolutions per minute, 

27tNPl 

hp = — — = o.oooiqoaNP/. . . . (8iV 

33 000 v J 

It is seen that this method is independent of the radius of the 
pulley, which may be of any convenient size ; for a small motor 
the brake may be clamped directly upon the shaft, but for a 
large one a pulley of considerable size is needed, and a special 
arrangement of levers is used instead of a cord. 

The efficiency of the motor is now found by dividing the 



294 MEASUREMENT 'OF WATER POWER. [Chap. X. 

effective work by the theoretic energy, or the effective power 
by the theoretic power ; thus : 

k hp , 11PI 

e = K = 77? = 6 - 2S iwr • ; • ■ < 82 > 

This same formula applies if the number of revolutions be per 

minute, provided that W be the weight of water which flows 
through the wheel per minute. 

The power measured by the friction brake is that delivered 
at the circumference of the pulley, and does not include that 
power which is required to overcome the friction of the shaft 
upon its bearings. The shaft or axis of every water wheel 
must have at least two bearings, the friction of which consumes 
probably about 2 or 3 per cent of the power. The hydraulic 
efficiency of the wheel, regarded as a user of water, is hence 2 
or 3 per cent greater than the value of e as given by (82). 

There are in use various forms and varieties of the friction 
brake, but they all act upon the principle and in the manner 
above described. For large wheels they are made of iron, and 
almost completely encircle the pulley; while a special arrange- 
ment of levers is used to lift the large weight Pr If the 
work transformed into friction be large, both the brake and the 
pulley may become very hot, to prevent which a stream of cool 
water is allowed to flow upon them. To insure steadiness of 
motion it is well that the surface of the pulley should be lubri- 
cated, which for a wooden brake is well done by the use of 
soap. 

Prob. 150. Find the power and efficiency of a motor when 
the theoretic energy is 1.38 horse-power, which makes 670 revo- 
lutions per minute, the weight on the brake being 2 pounds 
14 ounces and its lever arm 1.33 feet. 

* A paper by Thurston in Transactions of American Society of Mechanical 
Engineers 1SS6, vol. viii. , gives detailed descriptions and illustrations of the 
testing apparatus at Holyoke, Mass. 



Art. 125.] TEST OF A SMALL MOTOR. 295 

Article 125. Test of a Small Motor. 

The following description and notes of a test of the 6-inch 
Eureka turbine in the hydraulic laboratory of Lehigh Univer- 
sity may serve to exemplify the preceding method of determin- 
ing the effective power and efficiency. The water was deliv-\ 
ered over a weir from which it ran into a vertical penstock 
15.98 square feet in horizontal cross-section. This plan of hav- 
jng the weir above the wheel is in general not a good one, since 
it is then difficult to maintain a constant head in the penstock ; 
and it was adopted in this case on account of the lack of room 
below the wheel, and for other reasons which need not here be 
explained, as they are not related to the question in hand. The 
weir is briefly described in Art. 52, and the depths on its crest 
were determined by a hook gauge reading to thousandths of a 
foot. When a constant head is maintained in the penstock 
the quantity of water flowing through the wheel is the same as 
that passing the weir ; if, however, the head in the penstock 
falls x feet per minute, the flow Q through the wheel in cubic 
feet per minute is 

Q = 6o?-f- 15.98*, 

in which q is the flow per second through the weir, as com- 
puted by the methods of Chapter V. As the supply of water 
was very limited the wheel could not be run to its full capacity. 
There was no leakage from the penstock, and the slight leak- 
age through the gate of the turbine is properly included in the 
value of q, since it assists in running the wheel. 

The level of water in the penstock above the wheel was 
read upon a head gauge consisting of a glass tube behind which 
a graduated scale was fixed, the zero of which was a little above 
the water level in the tail race. The latter level was read upon 
a fixed graduated scale having its zero in the same horizontal 
plane as the first ; these readings were hence essentially nega- 



296 



MEASUREMENT OF WATER POWER. [Chap. X. 



tive. The head upon the wheel is then found by adding the 
readings of the two gauges. 

The vertical shaft of the turbine, being about 15 feet long, 
was supported by four bearings, and to a small pulley upon its 
upper end was attached the friction dynamometer, as described 
in the last article. The number of revolutions was read from 
a counter placed in the top of this shaft. The observations 
were taken at minute intervals, electric bells giving the signals, 
so that precisely at the beginning of each minute simultaneous 
readings were taken by observers at the weir, at the head gauge, 
at the tail gauge, and at the counter, the operator at the brake 
continually keeping it in equilibrium with the resisting weight 
in the scale pan by slightly tightening and loosening the nuts 
as required. The following table gives the notes of four sets, 

TABLE XXIII. TEST OF A 6-INCH EUREKA TURBINE. 



Time on 


Depth 


Penstock 


Tail-race 


Revolu- 


Weight 




April 13, 


on Weir 


Gauge. 


Gauge. 


tions in 


on 


Remarks. 


1888. 


Crest. 






One 


Brake. 






Feet. 


Feet. 


Feet. 


Minute. 


Pounds. 




3 h i7 OT 


0.2SS 


II.25 


— 0.2I 


635 


2-5 




18 


O.2S9 


II. 17 


0.20 


625 




Length of weir, 


19 


O.2S9 


II. 13 


0.2I 




i e 


b = 1.909 feet. 


20 


0.2SS 


II . 10 


0.2I 


635 


" 


Length of lever- 
arm on brake. 

/ = 1. 431 feet. 


oh OO™ 1 


0.287 


10. Si 


— O.20 


535 


3.0 


23 


O.2S7 


10.69 


0.20 




" 




24 


O.2S7 


10.62 


0.2I 


540 


« 


Gate of wheel fths 
open during all ex- 








535 




25 


O.2S6 


io.57 


0.2I 






periments. 


3 h 2; m 


o.2S3 


10.64 


— O.23 


600 


2-5 


Temperature of 
the water not taken. 


23 


0.2SS 


10. 72 


O. 22 


600 






29 ' 


0.291 


10. So 


0.2I 


615 






SO 


0.290 


10.90 


0.20 








3 h .^n 


0.290 


10.72 


— 0.20 




3-5 










445 






33 


0.291 


10.69 


0.20 


440 






34 


0.291 


10.66 


0.20 


440 






35 


0.292 


10.64 


0.20 









Art. 125.] 



TEST OF A SMALL MOTOR. 



297 



each lasting three minutes, the weight in the scale pan being 
different in each. In the intervals between the sets the wheel 
was kept running and observations were regularly taken ; but 
they are not used, owing to the disturbance of the permanent 
motion consequent upon changing the weight in the scale-pan. 

The following table gives the results of the computations 
made from the above notes for each minute interval. The 
second column is derived from formula (33) of Art. 53, using 
TABLE XXIV. RESULTS OF TEST OF A 6-INCH TURBINE. 




the coefficient corresponding to the given length of weir and 
depth on crest. The third column is obtained by taking the 
differences of the observed readings of the penstock head 
gauge. The fourth column is the value of Q found as above 
explained. The fifth column is the mean head h on the wheel 
during the minute, as derived from the observed readings of 
head and tail gauge. The sixth column is found by formula 
(80), using for W its value -fowQ, in which w is taken at 62.4 



298 MEASUREMENT OE WATER POWER. [Chap. X. 

pounds per cubic foot. The seventh column is computed from 
formula (81)' ; and the last column is found by dividing the 
numbers in the seventh by those in the sixth column. 

These results show that the mean efficiency of the wheeL 
and shaft under the conditions stated was about 35 per cent; 
also, that the efficiency in the second set is the highest and 
that in the first is the lowest. The following recapitulation of 
the means for each set show that the reason for the variation 
in efficiency is the variation in speed, and it is to be concluded 





N. 


h. 


Q. 


e. 


1st set, 


632 


II.36 


59-31 


33.8 


3d set, 


605 


IO.98 


57.27 


34-6 


2d set, 


536 


IO.87 


59-33 


36.O 


4th set, 


44I 


IO.87 


59.68 


34-3 



that with a head of about 1 1 feet this wheel at three-fourths 
gate has a maximum efficiency of 36.0 per cent when running 
at about 535 revolutions per minute. If four points be plotted, 
taking the values of N as abscissas and those of e as ordinates, 
and a curve be drawn through them, it will be seen that quite 
material variations in the speed may occur without sensibly 
affecting the efficiency; thus N may range from 475 to 575 
revolutions per minute without making e lower than 0.35. 

Prob. 151. Compute, using four-figure logarithms, the re- 
sults in the last three columns of the above table. 

Article 126. Lowell and Holyoke Tests. 

The work of Francis on the experiments made by him at 
Lowell will always be a classic in American hydraulic litera- 
ture, for the methods therein developed for measuring the 
theoretic power of a water-fall, and the effective power utilized 
by the wheel, are models of careful and precise experimenta- 
tion." In determining the speed of the wheel he used a method 

* Lowell Hydraulic Experiments, 1st Edition, 1S55 ; 41b. 1SS3. 



Art. 126.] LOWELL AND HOLYOKE TESTS. 299 

somewhat different from that above explained, namely, the 
counter attached to the shaft was connected with a bell which 
struck at the completion of every 50 revolutions ; the observer 
at the counter had then only to keep his eye upon the watch, 
and to note the time at certain designated intervals — say at 
every sixth stroke of the bell. The number of revolutions per 
second was then obtained by dividing the number of revolu- 
tions in the interval by the number of seconds, as determined 
by the watch. This method requires a stop-watch in order to 
do good work, unless the observer be very experienced, as an 
error of one second in an interval of one minute amounts to 
1.7 per cent. 

The Holyoke Water Power Company has a permanent 
flume for testing turbines arranged with a weir which can be 
varied up to lengths of 20 feet, so as to test the largest wheels 
which are constructed. As the expense of fitting up the ap- 
paratus for testing a large turbine at the place where it is to 
be used is often great, it is sometimes required in contracts 
that the wheel shall be sent to Holyoke to be tested, that be- 
ing the only place in the United States where a special outfit 
for such work exists. The wheel is mounted in the testing 
flume, and there, by the methods explained in the preceding 
articles, it is run at different speeds in order to determine the 
speed which gives the maximum efficiency as well as the effec- 
tive power developed at each speed. As the efficiency of a 
turbine varies greatly with the position of the gate which ad- 
mits the water to it, tests are made with the gate fully opened 
and at various partial openings. The results thus obtained are 
not only valuable in furnishing full information concerning the 
effective power and efficiency of the wheel, but they also con- 
vert the turbine into a water meter, so that when running under 
the same head as during the tests the quantity of water which 
passes through it can at any time be approximately ascer- 
tained. 



;oo 



MEASUREMEXT OE WATER POWER. 



[Chap. X. 



The following table gives a report of the Holyoke W ater 
Company of the test of an 8o-inch outward-flow BOYDEX tur- 
bine made September 22, 1885.^ The headings of the several 



TABLE XXV. TEST OF AN So-INCH BOYDEX TURBINE. 





Proportional Part 
of 








Quantity 








the full 


Number 




Discharge 

of the 


Head 


Dura- 


Revolu- 


of 
Water 


Power 


Efficien- 1 


of the 


the full 


Wheel ; 


acting 


tion of 


tions 


dis- 


developed 


cy of 


Experi- 


Opening 
of the 


being the 


on the 


the Ex- 


of the 


• charged 


bv the 


the 


ment. 


Discharge 
at full 


Wheel. 


peri- 


Wheel. 


by the 


Wheel. 


Wheel. 




Speed- 




ment. 




Wheel. 








gate. 


Gate 

when giv- 
ing best 








Cub. Ft. 
per 










Efficiency 


Feet. 


Min. 


Per Min. 


Second. 


h. p. 


Per Cent 


21 


I. OOO 


O.992 


17.16 


5 


63-50 


117. 01 


172.57 


75-85 


20 


i < 


1. 000 


17.27 


5 


70.OO 


118.37 


177-41 


76.60 


19 


" 


1.00S 


17-33 


3 


75.OO 


H9-53 


17S.63 


76.II 


18 


' ' 


1.020 


17-34 


3 


80.OO 


121 . 15 


178.32 


74-92 


17 


' ' 


1.036 


17.21 


2 


86. 00 


122.41 


I7S.57 


74- 81 


16 


" 


1.056 


17.21 


5 


93.20 


124.74 


176.44 


72.54 


15 


' ' 


1.082 


17.19 


3 


100.00 


127-73 


167.94 


67-5I 


14 


0.753 


0.923 


17.26 


4 


65.00 


109.22 


145.56 


69.69 


13 


' ' 


0.931 


I 7 • 35 


4 


71 .00 


110.42 


151.76 


69.91 


12 


" 


0.944 


i7o3 


3 


77.17 


in. 94 


153.16 


69. 68 


II 


' ' 


0.957 


17-34 


3 


82.83 


"3-52 


151-75 


68. 04 


10 


" 


0.9S6 


17-34 


3 


93-33 


116. 9S 


151-04 


65.72 


9 


' ' 


0.999 


17.27 


4 


97-75 


1 1 8. 24 


140.29 


60.63 


8 


' ' 


1. 018 


17-23 


4 


104.50 


120.36 


130.53 


55-68 


33 


O.609 


O.S49 


17.64 


4 


65.00 


101.60 


130.99 


64-51 


32 


" 


0.861 


17-57 


3 


71 .00 


102.78 


130.08 


63.58 


3i 


' ' 


0.S76 


17-53 


4 


78. 00 


104.45 


12S.61 


62.00 


30 


' ' 


0.892 


17-45 


4 


84.75 


106.19 


124.22 


59- ] 6 


7 


O.43S 


0.706 


17.68 


3 


64.00 


84.^6 


S4.03 


49.61 


6 


. i 


0.716 


17.69 


4 


69.25 


85-74 


82. 47 


47 99 


5 


1 ' 


0.723 


17.69 


4 


74-75 


S6.57 


79. 89 


46.04 


4 


" 


0.731 


17.66 


4 


79-S7 


87-54 


75.60 


43- 16 


3 


' ' 


0.746 


17.62 


4 


S6.50 


S9.23 


68. 67 


38.55 


2 


1 ' 


0.762 


17.64 


3 


94-33 


91.12 


57-6i 


31.63 


1 


" 


0.773 


17.61 


4 


100. 50 


92.36 


46.03 


24. 9* 


27 


O.310 


0.555 


1S.03 


4 


61.75 


67. 11 


43-37 


31 .63 


26 


" 


0.570 


1S.01 


4 


69.25 


68. S7 


40. 1 S 


28.59 


25 


' ' 


0.5S4 


1S.02 


3 


77.00 


70 59 


35-27 


24.47 


2S 


' ' 


0.597 


IS. 23 


3 


85. 00 


72.62 


2&.55 


19-03 


29 


'' 


0.608 


IS. 13 


3 


90.67 


75 -74 


19.38 


12.79 


24 


0.200 


0.401 


iS. 17 


3 


54-33 


48 .68 


18.25 


18.21 


23 


' ' 


0.409 


i3.ii 


3 


61.67 


49-52 


15 .06 


14.53 


22 




0.413 


1S.07 


4 


66 50 


50.03 


12. 18 


ii.5o 



* Kindly furnished by Mr. Clemens Herschel. Hydraulic Engineer of the 
Holyoke Water Power Company. 



Art. 126.] LOWELL AND HOLYOKE TESTS. 3OI 

columns sufficiently designate their meaning, and only a few 
words need be added explanatory of the method by which 
they were obtained. The numbers in the second column were 
found by actual measurement of the clear space beneath the 
gate, the space at full gate being called unity ; those in the 
fourth column are derived from the head and tail race gauges; 
those in the sixth column by dividing the total number of revo- 
lutions during the experiment by its length in minutes ; those 
in the seventh by the measurement of the water over the weir ; 
those in the eighth from the friction dynamometer by the use 
of formula (81)' ; and those in the last column were computed 
by (82). The quantities in the third column result from the 
division of those in the seventh by 118.37, tnat being the dis- 
charge at full gate for maximum efficiency ; it is seen from 
these that the discharge depends not only upon the head, but 
on the velocity of the wheel, and that it always increases when 
the speed increases.* 

The following data regarding the dimensions of this wheel 
are here also noted, as it may be necessary to refer to it again 
when the subject of turbines is discussed : 

Outer radius of wheel r x = 3.3167 feet ; 

Inner radius of wheel r = 2.6630 feet; 

Outer radius of guide case r = 2.591 1 feet; 

Outer depth of buckets d x = 0.722 feet; 

Inner depth of buckets d = 0.741 feet; 

Outer area of buckets a x = 4.61 square feet; 

Inner area of buckets a = 12.12 square feet; 

Outer area of guide orifices a = 4.76 square feet ; 

Exit angle of buckets (3 = 13.5 degrees; 

Entrance angle of buckets = 90 degrees ; 

Entrance angle of guides a — 24 degrees ; 
Number of buckets, 52. Number of guides, 32. 

* For further examples of tests at Holyoke see a paper by Thurston in 
Transactions American Society Mechanical Engineers, vol. viii. p. 359. 



302 MEASUREMEXT OF WATER POWER. [Chap. X. 

Prob. 152. In experiment 22 on the outward-flow turbine 
only about 12 per cent of the theoretic power is utilized. How 
is the remaining 88 per cent expended? 

Article 127. Water Power. 

In 1880 there was employed in the United States a total of 
3 410 837 horse-power, of which about 36 per cent was derived 
from water and about 64 per cent from steam. It has been 
estimated that the rivers of the United States can furnish 
about 200000000 horse-powers, so that the possibilities for the 
future are almost unlimited, and when coal becomes high in 
price water is sure to take the place of steam. 

Water-power is often sold by what is called the " mill 
power," which may be roughly supposed to be such a quantity 
as the average mill requires, but which in any particular case 
must be defined by a certain quantity of water under a given 
head. Thus at Lowell the mill power is 30 cubic feet per 
second under a head of 25 feet, which is equivalent to 85.2 
theoretic horse-power. At Minneapolis it is 30 cubic feet per 
second under 22 feet head, or 75 theoretic horse-power. At 
Holyoke it is 38 cubic feet per second under 20 feet head, or 
86.4 theoretic horse-power. This seems an excellent way to 
measure power when it is to be sold or rented, as the head in 
any particular instance is not subject to much variation : or if 
so liable, arrangements must be adopted for keeping it nearly 
constant, in order that the machinery in the mill may be run at 
a tolerably uniform rate of speed. Thus nothing remains for 
the water company to measure except the water used by the 
consumer. The latter furnishes his own motor, and is hence 
interested in securing one of high efficiency, that he may derive 
the greatest power from the water for which he pays. The 
perfection of American turbines is undoubtedly largely due to 
this method of selling power, and the consequent desire of the 



Art. I27-] WATER POWER. 303 

mill owners to limit their expenditure of water. The turbine 
itself when tested and rated becomes a meter by which the 
company can at any time determine the quantity of water that 
passes through it. At Holyoke the cost of one mill power for 
16 hours a day is $300 per annum.* 

The available power of natural water-falls is very great, but 
it is probably exceeded by that which can be derived from the 
tides and waves of the ocean. Twice every day, under the 
attraction of the sun and moon, an immense weight of water is 
lifted, and it is theoretically possible to derive from this a 
power due to its weight and lift. Continually along every 
ocean beach the waves dash in roar and foam, and energy is 
wasted in heat which by some device might be utilized in 
power. The expense of deriving power from these sources is 
indeed greater than that of the water wheel under a natural 
fall, but the time may come when the profit will exceed the 
expense, and then it will certainly be done. Coal and wood 
and oil may become exhausted, but as long as the force of 
gravitation exists, and the ocean remains upon which it can act, 
heat, light, and power can be generated in quantity practically 
without limit. 

Prob. 153. Show that over 600 horse-power is wasted in 
heat for every square mile of ocean surface where the rise and 
fall of the tide is 3 feet. 

* Breckenridge, Journal of Engineering Society of Lehigh University, 
18S7, vol. ii. p. 34; an article giving a detailed account of the water power at 
Holyoke. 



304 DYNAMIC PRESSURE OF FLOWING WATER. [Chap.' XI. 



CHAPTER XL 
DYNAMIC PRESSURE OF FLOWING WATER. 

Article 128. Definitions and Principles. 

The pressures exerted by moving water against surfaces 
which change its direction or check its velocity are called 
dynamic, and they follow very different laws from those which 
govern the static pressures that have been discussed and used in 
the preceding chapters. A static pressure due to a certain head 
may cause a jet to issue from an orifice ; but this jet in imping- 
ing upon a surface may cause a dynamic pressure less than, equal 
to, or greater than that due to the head. A static pressure at a 
given point in a mass of water is exerted with equal intensity 
in all directions ; but a dynamic pressure is exerted in different 
directions with different intensities. In the following chapters 
the words static and dynamic will generally be prefixed to the 
word pressure, so that no intellectual confusion may result. 

The dynamic pressure exerted by a stream flowing with a 
given velocity against a surface at rest is evidently equal to 
that produced when the surface moves in still water with the 
same velocity. This principle was applied in Art. 109 in rating 
the current meter, whose vanes move under the impulse of the 
impinging water. The dynamic pressure exerted upon a body 
by a flowing stream hence depends upon the velocity of the 
stream and surface. 

The impulse of a jet or stream of water is the dynamic 
pressure which it is capable of producing in the direction of its 
motion when its velocity is entirely destroyed in that direction. 



Art. 128.] DEFINITIONS AND PRINCIPLES. 305 

This can be done by deflecting the jet normally sidewise by a 
fixed surface ; if the surface is smooth, so that no energy is lost 
in frictional resistances, the actual velocity remains unaltered, 
but the velocity in the original direction has been rendered 
null. In Art. 32 it is proved that the theoretic force of im- 
pulse of a stream of cross-section a and velocity v is 

v v v 2 

F = W — = zvq— = 2wa — , . . . . (83) 
g g 2g V DJ 

in which W and q are the weight and volume delivered per 
second, and w is the weight of one cubic unit of water. This 
equation shows that the dynamic pressure that may be pro- 
duced by impulse is equal to the static pressure due to twice 
the head corresponding to the velocity v. 
It would then be expected that if two 
equal orifices or tubes be placed ex- 
actly opposite, as in Fig. 81, and a loose 
plate be placed vertically against one 
of them, that the dynamic pressure 
upon the plate caused by the impulse 
of the jet issuing from A under the Fig. 81. 

head h would balance the static pressure caused by the head 
2I1. This conclusion has been confirmed by experiment, when 
the tube ^4 has a smooth inner surface and rounded inner edo-es 
so that its coefficient of discharge is unity. 

The reaction of a jet or stream is the backward dynamic pres- 
sure, in the line of its motion, which is exerted against a vessel out 
of which it issues, or against a surface away from which it moves. 
This is equal and opposite to the impulse, and the equation 
above given expresses its value and the laws which govern it. 

The expression for the reaction or impulse F given by (83) 
may be also proved as follows: The definition by which forces. 




306 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI. 

are compared with each other is, that forces are proportional 
to the accelerations which they can produce. The weight \h 
if allowed to fall acquires the acceleration g; the force F which 
can produce the acceleration v is hence related to W and g by 
the equation F -f- v = W -±- g. 

Impulse and reaction in a cross-section of a stream flowing 
with constant velocity and direction are forces which can be 
exerted, and hence like energy are potential. If the direction 
of the stream be changed by opposing obstacles, the impulse 
and reaction produce dynamic pressure ; if in making this 
change the absolute velocity is retarded, energy is converted 
into work. Impulse and reaction are of no practical value, 
except in so far as the resulting dynamic pressures may be 
utilized for the production of work. For this purpose water is 
made to impinge upon moving vanes, which alter both its direc- 
tion and velocity, thus producing a dynamic pressure P, which 
overcomes in each second an equal resisting force through the 
space //. The work done per second is then 

k = Pu. 

It is the object in designing a hydraulic motor to make this 
work as large as possible, and for this purpose the most advan- 
tageous values of P and ?/ are to be selected. 

The word impact, which is sometimes erroneously used to 
mean impulse or pressure, properly refers to those cases where 
energy is lost through changes of cross-section (Art. 68), or in 
eddies and foam, as when a jet impinges into water or upon a 
rough plane surface. When work is to be utilized, impact 
should be avoided as far as possible. 

Prob. 1 54. If a jet is one inch in diameter, how many gallons 
per second must it deliver in order that its impulse may be 100 
pounds? 




Art. 129.] EXPERIMENTS ON IMPULSE AND REACTION. Z°7 

Article 129. Experiments on Impulse and Reaction. 

In Fig. 82 is shown a simple device by which the dynamic 
pressure P exerted upon a surface by the impulse and reaction 
of a jet that glides over it can be directly weighed. It consists 
merely of a bent lever supported 
on a pivot at O, and having a plate 
A attached at lower end of the 
vertical arm upon which the stream 
impinges, while weights applied at ___ 
the end of the other arm meas- ^ 
ure the dynamic pressure produced 
by the impulse. By means of an 
apparatus of this nature, experiments have been made by 
BlDONE, WE1SBACH, and others, the results of which will now 
be stated. 

When the surface upon which the stream impinges is a 
plane normal to the direction of the stream, as shown at A, 
the dynamic pressure P closely agrees with that given by the 
theoretic formula for F in the last article, viz., 

v z? 

P= W- = 2wa — 
g 2 g 

being about 2 per cent greater according to BlDONE, and 
about 4 per cent less according to WEISBACH. The actual 
value of P was found to vary somewhat with the size of the 
plate, and with its distance from the end of the tube from which 
the jet issued. 

When the surface upon which the stream impinges is curved, 
as at B, or so arranged that the water is turned backward from 
the surface, the value of the dynamic pressure P was found to 
be always greater than the theoretic value, and that it increased 



JOS DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XL 



the greater the amount of backward inclination. When a 
complete reversal of the original direction of the water was 
obtained, as at C, it was found that P, as measured by the 
weights, was nearly double the value of that against the plane. 
This is explained by stating that as long as the direction of the 
flow is toward the surface the dynamic pressure of its impulse 
is exerted upon it ; when the water flows backward away from 
the surface the dynamic pressure of its reaction is also exerted 
upon it. The sum of these is 



v v* 

F = 2 W- = aw a — 



P = F 

which agrees with the results experimentally obtained. 



An experiment by MOROSI* shows clearly that the dynamic 
pressure against a surface may be increased still further by the 
device, shown in Fig. 83, where the stream is made to perform 
two complete reversals upon the surface. He found that in 
this case the value of the dynamic pressure was 
3.32 times as great as that against a plane, or 
P = 3.32 F, whereas theoretically the 3.32 should 
be 4. In this case, as in those preceding, the 
water in passing over the surface loses energy in 
friction and foam, so that its velocity is dimin- 
ished, and it should hence be expected that the experimental 
values of the dynamic pressures would be less than the theo- 
retic values, as in general they are found to be. 

While the experiments here briefly described thoroughly 
confirm the results of theory, they further show it is the change 
in direction of the velocity when in contact with the surface 
which produces the dynamic pressure. If the stream strikes 




Fig. 83. 



* Ruhlman's Hydromechanik (Hannover, 1S79), p. 5S6. 



Art. 129.] EXPERIMENTS ON IMPULSE AND REACTION. 309 




normally against a plane, the direction of its velocity is changed 
90 , and this is the same as the entire destruction of the 
velocity in its original direction, so that the dynamic pressure/^ 
should agree with the impulse F. This important principle of 
change in direction will be theoretically exemplified later. 

The dynamic pressure produced by the direct reaction of 
water when issuing from a vessel was meas- 
ured by Ewart with the apparatus shown 
in Fig. 84, which will be readily understood 
without a detailed description. The dis- 
cussion of these experiments made by WEIS- 
BACH* shows that the measured values of 
P were from 2 to 4 per cent less than the 
theoretic value F as given by (83), so that in FlG - 8 4- 

this case also theory and observation are in accordance. 

An experiment by UNWlN,f illustrated in Fig. 85, is very 
interesting, as it perhaps explains more clearly than formula 
{83) why it is that the dynamic pressure due to impulse is 
double the static pressure. Two ves-^ 



sels having converging tubes of equal 
size were placed so that the jet from 
A was directed exactly into B. The 
head in A was kept uniform at 2o|- 
inches, when it was found that the 
water in B continued to rise until a 
head of 18 inches was reached. 



Fig. 85. 

All the water admitted into A 
was thus lifted in B by the impulse of the jet, with a loss of 2-J 
inches of head, which was caused by foam and friction. If 
such losses could be entirely avoided, the water in B might be 
raised to the same level as that in A. In the case shown in 

* Theoretical Mechanics, Coxe's translation, p. 1004. 
f Encyclopaedia Britannica, 9th Edition, vol. xii. p. 467. 



310 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI. 

the figure where the water overflows from B, the impulse of 
the jet has not only to overcome the static pressure due to the 
head //, but also to furnish the dynamic pressure equivalent to 
a second head h in order to raise the water through that height. 
But the level in B can never rise higher than in A, for the ve- 
locity-head of the jet cannot be greater than that of the static 
head which generates it. 

Prob. 155. In Fig. 82 the diameter of the tube is 1 inch, 
and it delivers 0.3 cubic feet per second. Compute the theo- 
retic dynamic pressure against the plane. 

Article 130. Surfaces at Rest. 

Let a jet of water whose cross-section is a impinge in per- 
manent flow with the uniform velocity v upon a surface at rest. 
Let the surface be smooth, so that no resisting forces of fric- 
tion exist, and let the stream be prevented from spreading lat- 
erally. The water then passes over the surface, and leaves it 




with the original velocity v, producing upon it a dynamic pres- 
sure whose value depends upon its change of direction. At B 
in Fig. 86 the stream is deflected normal to its original direc- 
tion, and at D a complete reversal is effected, Let 6 be the 
angle between the initial and final directions, as shown. It is 
required to determine the dynamic pressure exerted upon the 
surface in the same direction as that of the jet. In Fig. S6, as 
in those that follow, the stream is supposed to lie in a horizon- 
tal plane, so that no acceleration or retardation of its velocity 
will be produced by gravity. 



Art. 130.] 



SURFACES AT REST. 



311 




—Jr. 



Fig. 87. 



The stream entering upon the surface exerts its impulse F 
in the same direction as 
that of its motion ; leaving 
the surface it exerts its reac- 
tion F in opposite direction 
to that of its motion. Let 
P be the dynamic pressure , F ; 
thus produced in the direc- 
tion of the initial motion, F 1 the component of the reaction F 
in the same direction. Then, if 6 be less than 90 , 

P~F-F 1 =F(i- cos 6) ; 

and if be greater than 90 , 

P=F + F 1 = F+Fcos(iSo° - 6) = F(i - cos 6). 

Both cases thus give the same result, and inserting for F its 
value as given by (83) ; 



(1 — cos 6)W 



g 



(84) 



which is the formula for the dynamic pressure in the direction 
of the impinging jet. If in this = o°, the stream glides along 
the surface without changing its direction, and P becomes zero ; 
if 6 is 90 , the dynamic pressure is 

P = F= W-\ 
g 

and if 6 becomes 180 a complete reversal of direction is ob- 
tained, and 

v 



P=2F=2W- 

g 

These theoretic conclusions agree with the experimental results 
described in the last article. 

The resultant dynamic pressure exerted upon the surface is 
found by combining by the parallelogram of forces the impulse 




312 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI. 

F and the equal reaction F. In Fig. 87 it is seen that this 
resultant bisects the angle 180 — 6, and that its value is 

P' = 2Fcosi(i8o— d) = 2sinid.W-. 

<*> 

It is usually, however, more important to ascertain the pres- 
sure in a given direction than the resultant. 
ft This can be found by taking the component 
\ of the resultant in that direction, or by tak-. 

/ ing the algebraic sum of the components of 
•it ... . 

\ the initial impulse and the final reaction. 

To find the dynamic pressure P in a di- 
rection which makes an angle a with the 
entering and the angle 6 with the depart- 
ing stream, the components in that direction are 

P 1 = F cos a, P n = F cos ; 

and the algebraic sum of these is 

P = F(cos a — cos 0) = (cos a — cos 6) W— . . (84)' 

This becomes equal to F when a = o and = 90°, as at B in 
Fig. 86, and also when a = 90 and 6 = 180 . When a = o° 
and = 180 the entering and departing streams are parallel, 
as at D in Fig. 86, so that the value of P is 2F, which in this 
case is the same as the resultant pressure. 

The formulas here deduced are entirely independent of the 
form of the surface, and of the angle with which the jet enters 
upon it. It is clear, however, if, as in the planes in Fig. 86, this 
angle is such as to allow shock to occur, that foam and changes 
in cross-section may result which will cause energy to be dissi- 
pated in heat. These losses will diminish the velocity v and 
decrease the theoretic dynamic pressure. These effects cannot 
be formulated, but it is a general principle, which, is confirmed 



Art. 131.] CURVED PIPES AND CHANNELS. 3 T 3 

by experiment, that they may be largely avoided by allowing 
the jet to impinge tangentially upon the surface. 

In all the foregoing formulas the weight W which im- 
pinges upon the surface per second is the same as that which 
issues from the orifice or nozzle that supplies the stream, and 
its value is 

W •=■ wq = waVo 

To find W it is hence necessary to determine the discharge q 
by the methods explained in the preceding chapters, or to 
measure #, the area of the cross-section of the stream, and to 
ascertain by some method the mean velocity v. 

Prob. 156. If F is 10 pounds, a = o°, and = 6o°, show 
that the pressure parallel to the direction of the jet is 5 
pounds, that the pressure normal to that direction is 8.66 
pounds, and that the resultant dynamic pressure is 10 pounds. 



Article 131. Curved Pipes and Channels. 

The dynamic pressures discussed in the preceding article 
have been those caused by jets, or isolated streams, of water. 
There is now to be considered the case of dynamic pressures 
caused by streams flowing in pipes, conduits, or channels of any 
kind ; these are sometimes called limited or bounded streams, 
the boundary being the surface whose cross-section is the wetted 
perimeter. When such a stream is straight and of uniform 
section, and all its filaments move with the same velocity v, the 
impulse, or the pressure which it can produce, is the quantity 
F given by the general expression in Art. 128; under these 
conditions it exerts no dynamic pressure, but if a body be im- 
mersed and held stationary, pressure is produced upon it. If 
its direction changes in an elbow or bend, pressure upon the 



314 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI. 

bounding surface is produced ; if its cross-section increases 
or decreases, pressure is developed or absorbed. 

The resultant dynamic pressure P' upon a curved pipe 
which runs full of water with the uniform velocity v depends 
upon the angle 6 between the initial and final directions, and 




Fig. 89. 

must be the same as that produced upon a surface by an im- 
pinging jet which passes over it without change in velocity. 
The formula of Art. 130 then directly applies, and 

P' = 2 sin \B . F = 2 sin Id . W- . 

* cr 

If 6 — 0°, there is no bend, and P r = o; if # — 180 , the direc- 
tion of flow is reversed, and P' = 2F. If the direction is twice 
reversed, as at C in Fig. 89, the value of 6 is 360 , and the re- 
sultant is the initial impulse F minus the final reaction F, or 
simply zero ; in this case, however, there may be a couple 
which tends to twist the pipe, unless the impulse .and reaction 
lie in the same line. 



The total dynamic pressure exerted upon the curved pipe 
may be found by taking the sum of all the elementary radial 
pressures. For this purpose let the pipe at A in Fig. 89 have 
the length dl and let 6 be nearly o°, so that its value is the ele- 
mentary angle Sd. Then in the above formula P' becomes the 
elementary radial pressure SP 1 , and 

SP, = 2 sin i$6 . F = FS6. 



Art. 131.] 



CURVED PIPES AND CHANNELS. 



315 



Now for a circular curve whose radius is R, the value of 61 is 
RS6 ; and accordingly the elementary radial pressure for that 
case is expressed by the dffferential equation 

The total radial pressure P l upon a circular curve whose 
length is / is the integral of this equation between the limits o 
and P 1 for the first member and o and / for the second, or 

_ Fl I v* 

P,=-— = 2wa 75 — . 

1 R R 2g 

This dynamic pressure does no work and offers no direct re- 
sistance in the direction of the flow ; but in being transmitted 
through the water to the outer side of the pipe it causes cross- 
currents which consume energy. This expression for radial 
pressure is the same as that given by the theory of centrifugal 
force. It is not strictly exact unless all the filaments have the 
same velocity v, which in a curved pipe is probably never 
the case. 



The same reasoning applies approximately to the curves 
of conduits, canals, and rivers. In any length / there exists 
a radial dynamic pressure P 1 , acting toward the outer bank 
and causing; currents in that 



direction, which, in connec- 
tion with the greater ve- 
locity that naturally there 
exists, tends to deepen the 
channel on that side. This 
may help to explain the rea- 
son why rivers run in winding 
courses. At first the curve 




Fig. 90. 



may be slight, but the radial flow due to the dynamic pres- 



3l6 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI. 

sure causes the outer bank to scour away ; this increases the 
velocity v u and decreases v 1 (Fig. 901, and this in turn hastens 
the scour on the outer and allows material to be deposited on 
the inner side. Thus the process continues until a state of 
permanency is reached, and then the existing forces tend to 
maintain the curve. The cross-currents which the radial pres- 
sure produces have been actually seen in experiments devised 
by THOMSON,* and it is found that they move in the manner 
shown in Fig. 90, the motion toward the outer bank being in 
the upper part of the section, while along the wetted perimeter 
they flow toward the inner bank. 

The elevation of the water on the outer bank of a bend 
is higher than on the inner. This is only a practical conse- 
quence of the radial dynamic pressure, as in straight streams 
it is also seen that the water surface is curved, the highest 
elevation being where the velocity is greatest. In this case 
cross-currents are also observed which move near the surface 
toward the centre of the stream, and near the bottom toward 
the banks, their motion being due to the disturbance of the 
static pressure consequent upon the varying water level. 

Prob. 157. Why is it that streams of slight slope have the 
most winding courses ? 



Article 132. Immersed Bodies. 

When a body is immersed in a flowing stream, or when it 
is moved in still water, so that filaments are caused to change 
their direction, a dynamic pressure is exerted or overcome. 
The theoretic determination of the intensity of this pressure is 
difficult, if not impossible, and will not be here attempted: 

* Proceedings Royal Society, 1S77, P- 356. 



Art. 132. J 



IMMERSED BODIES. 



317 




in fact, experiment alone can furnish reliable conclusions. It 
is, however, to 
be inferred from 
what has pre- 
ceded, that the 
dynamic pressure Fig. 91. 

in the direction of the motion is proportional to the force of 
impulse of a stream whose cross-section is the same as that of 
the body, or 

P = m . wa — , 

in which m is a number depending upon the length and shape 
of the immersed portion, and whose value is 2 for a jet imping- 
ing normally upon a plane. 

Experiments made upon small plates held normally to the 
direction of the flow show that the value of m lies between 
1.25 and 1.75, the best determinations being near 1.4 and 1.5. 
It is to be expected that the dynamic pressure on a plate in a 
stream would be less than that due to the impulse of a jet of 
the same cross-section, as the filaments of water near the outer 
edges are crowded sideways, and hence do not impinge with 
full normal effect, and the above results confirm this supposi- 
tion. The few experiments on record were made with small 
plates, mostly less than 2 square feet area, and they seem to in- 
dicate that m is greater for large surfaces than for small ones. 

The determination of the dynamic pressure upon the end of 
a cylinder, as at B in Fig. 91, is difficult because of the resist- 
ing friction of the sides ; but it is well ascertained to be less 
than that upon a plane of the same area, and within certain 
limits to decrease with the length. For a conical or wedge- 
shaped body the dynamic pressure is less than that upon the 
cylinder, and it is found that its intensity is much modified 
by the shape of the rear surface. 



318 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI, 

When a body is so formed as to gradually deflect the fila- 
ments of water in front, and to allow them to gradually close in 
again upon the rear, the impulse of the front filaments upon the 
body is balanced by the reaction of those in the rear, so that the 
resultant dynamic pressure is zero. The forms of boats and 
ships should be made so as to secure this result, and then the 
propelling force has only to overcome the frictional resistance 
of the surface upon the water. 

The dynamic pressure produced by the impulse of ocean 
waves striking upon piers or lighthouses is often very great. 
The experiments of STEVENSON* on Skerryvore Island, where 
the waves probably acted with greater force than usual, showed 
that during the summer months the mean dynamic pressure 
per square foot was about 600 pounds, and during the winter 
months about 2100 pounds, the maximum observed value be- 
ing 6100 pounds. At the Bell Rock lighthouse the greatest 
value observed was about 3000 pounds per square foot. The 
observations were made by allowing the waves to impinge 
upon a circular plate about 6 inches in diameter, and the pres- 
sure produced was registered by the compression of a spring. 

Prob. 158. Compute the probable dynamic pressure upon a 
surface one foot square when immersed in a current whose 
velocity is 8 feet per second, the direction of the current being 
normal to the surface. 

Article 133. Moving Vanes. 

A vane is a plane or curved surface which moves in a given 
direction under the dynamic pressure exerted by an impinging 
jet or stream. The direction of the motion of the vane de- 
pends upon the conditions of its construction ; for example, 
the vanes of a water wheel can only move in a circumference 
around its axis. The simplest case for consideration, however. 



Rankine's Civil Engineering, p. 756. 



Art. 133.] MOVING VANES. 319 

is that where the motion is in a straight line, and this alone 
will be considered in this article. The plane of the stream and 
vane is to be taken as horizontal, so that no direct action of 
gravity can influence the action of the jet. 

Let a jet with the velocity v impinge upon a vane which 
moves in the same direction with the velocity u, and let the 
velocity of the jet relative to the surface y 
at the point of exit make an angle ft with 
the reverse direction of u, as shown in 
Fig. 92. The velocity of the stream rela- 
tive to the surface is v — u, and the dy- y \(®/ 
namic pressure is the same as if the sur- v-u" '~ Vi 
face were at rest, and the stream moving Fig. 92. 
with the absolute velocity v — it. Hence formula (84) directly 
applies, replacing v by v — u, and by — /?, and 

P = (i Jf-costyW 7 ^-^. 

In this formula W is not the weight of the water which issues 
from the nozzle, but that which strikes and leaves the vane, or 
W = wa(v — u) ; for under the condition here supposed the 
vane moves continually away from the nozzle, and hence does 
not receive all the water which it delivers. 

Another method of deducing the last equation is as follows : 
At the point of exit let lines be drawn representing the veloci- 
ties v — u and u ; then completing the parallelogram, the line 
z\ is the absolute velocity of the departing jet (Art. 33). Let 
6 be the angle which z\ makes with the direction of u, and (3 
as before the ansde between v — u and the reverse direction of 

o « 

u. Then the dynamic pressure is that due to the absolute im- 
pulse of the entering and departing streams ; the former of 

W W 

these has the value —v and the latter the value — v. cos 6. 
g g 



320 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XL 



Hence it is expressed by the formula 

- W 



g 



(v — v 1 cos 



But from the triangle between v x and u 

v 1 cos 6 = u — (v — u) cos /?. 
Inserting this, the value of the dynamic pressure is 

which is the same as that found before. If. j3 = 180 there is 
no pressure, and if /3 = o° the stream is completely reversed, 
and P attains its maximum value. The dynamic pressure may 
be exerted with different intensities upon different parts of the 
vane, but its total value in the direction of the motion is that 
given by the formula. 

Usually the direction of the motion is not the same as that 
of the jet. This case is shown in Fig. 93, where the arrow 
marked F designates the direction of the impinging jet, and 
that marked P the direction of the motion of the vane, a be- 
ing the angle between them. 
The jet having the velocity v 
impinges upon the vane at A, 
and in passing over it exerts 
a dynamic pressure P which 
causes it to move with the 
velocity u. At A let lines be 
drawn representing the inten- 
Fig. 93. sities and directions of v and 

u, and let the parallelogram of velocities be formed as shown ; 
the line V then represents the velocity of the water relative to 
the vane at A. The stream passes over the surface and leaves 
it at B with the same relative velocity V, if not retarded by 




Art. 133.] MOVING VANES. 32 1 

friction or foam. At B let lines be drawn to represent u and 
V f and let /3 be the angle which V makes with the reverse 
direction of u ; let the parallelogram be completed, giving v x 
for the absolute velocity of the departing water, and let 6 be 
the angle which it makes with u. The total pressure in the 
direction of the motion is now to be regarded as that caused 
by the components in that direction of the initial and the final 
impulse of the water. The impulse of the stream before strik- 
er 
ing the vane is — v, and its component in the direction of the 

w 

motion is — v cos a. That of the stream as it leaves the vane 
g 

W 
is — v x , and its component upon the direction of the motion is 

W 

— z\ cos 6. The difference of these components is the total 

pressure in the given direction, or 

W 

P = — (v cos a — z\ cos .0) (85) 

<5 

This is a general formula for the pressure in any given direc- 
tion upon a vane moving in a straight line. If the surface be 
at rest v 1 equals v, and it agrees with the result deduced in 
Art. 130. 

If it be required to find the numerical value of P, the given 
data are the velocities v and u, and the angles a and /3. The 
term v 1 cos 6 is hence to be expressed in terms of these quanti- 
ties. From the triangle at B between z\ and u 

v x cos Q ■=. u — V cos /?. 

Substituting this, the formula becomes 

W 

P— — {v cos a — u -\- V cos /?), . . . (85)' 



322 DYNAMIC PRESSURE OF FIOWING WATER. [Chap. XI. 

which is often a more convenient form for discussion. The 
value of V is found from the triangle at A between u and v, thus : 

V' 1 = u~ 4- v 1 — 211V cos ol ; 

and hence the dynamic pressure P is fully determined in terms 
of the given data. 

In order that the stream may enter tangentially upon the 
vane, and thus prevent foam, the curve of the vane at A should 
be tangent to the direction of V. This direction can be found 
by expressing the angle in terms of the given angle a. Thus 
from the relation between the sides and angles of the triangle 
included between u, v, and V there is found 

sin (0 —a) u 
sin v ' 

which reduces to the form 

cot = cot a 



v sin a 

from which can be computed when u, v, and a are given. If 
the angle made by the vane with the direction of the motion 
be greater or less than some loss due to impact will result. 

Prob. 159. What does z\ represent in the parallelogram 
drawn at B in Fig. 93 ? Express its value in terms of /?, u, 
andF. 



Article 134. Work derived from Moving Vanes. 

The work imparted to a moving vane by the energy of the 
impinging water is equal to the product of the dynamic pres- 
sure P, which is exerted in the direction of the motion and the 
space through which it moves. If u be the space described in 
one second, or the velocity of the vane, the work per second is 

k = Pt. 



Art. 134.] WORK DERIVED FROM MOVING VANES. 323 

This expression is now to be discussed in order to determine 
the value of tc which makes k a maximum. 

When the vane moves in a straight line in the same direc- 
tion as the impinging jet and the water enters it tangentially, 
as shown in Fig. 87, the work imparted is found by inserting for 
P its value from (84), whence 

**=(! + cos /J) W^> = (I + cos fiwa ( — -^ . 

The value of u which renders k a maximum is obtained by 

equating to zero the derivative of k with respect to u, or 

Sk zva 

— = (1 _|_ cos /?) — {v* - Atvu + 321*) = o, 
<?> 

from which the value of u is 

u = \v ; 

and inserting this, the maximum work is found to be 

3 
k= 8(1 +cos/?) — — . 



The theoretic energy of the impinging jet is 

K= W-- = wa—, 

ig zg 

and accordingly the efficiency of the vane is (Art. 31) 

k 8 "' 
e = -g = --(i +cos^). 

If fi = 180 , the jet glides along the vane without producing 
work and e = o ; if f$ = 90 , the water departs from the vane 
normal to its original direction and e = -g- 8 T ; if (3 = o, the direc- 
tion of the stream is reversed and e = ^ . 

It appears from the above that the greatest efficiency which 
can be obtained by a vane moving in a straight line under the 
impulse of a jet of water is |4 ; that is, the effective work is 
only about 59 per cent of the theoretic energy attainable. 



324 DYXAMIC PRESSURE OF FLOWIXG. WATER. [Chap. XI. 

This is due to two causes : first, the quantity of water which 
reaches and leaves the vane per second is less than that fur- 
nished by the nozzle or mouthpiece from which the water 
issues ; and secondly, the water leaving the wane still has an ab- 
solute velocity of \v. A vane moving in a straight line is 
therefore a poor arrangement for utilizing energy, and it will 
also be seen upon reflection that it would be impossible to con- 
struct a motor in which a vane would move continually in the 
same direction away from a fixed nozzle. The above discussion 
therefore gives but a rude approximation to the results ob- 
tainable under practical conditions. It shows truly, however, 
that the efficiency of a jet which is deflected normally from its 
path is but one half of that obtainable when a complete reversal 
of direction is made. 

Water wheels which act under the impulse of a jet consist 
of a series of vanes arranged around a circumference which by 
the motion are brought in succession before the jet. In this 
case the quantity of water which leaves the wheel per second 
is the same as that which enters it, so that IF does not depend 
on the velocity of the vanes, as in the preceding case, but is a 
constant Avhose value is wq, where q is the quantity furnished 
per second. An approximate estimate of the efficiency of a. 
series of such vanes can be made by considering a single vane 
and taking W as a constant. The water is supposed to im- 
pinge tangentially, and the vane to move in the same direction 
as the jet [Fig. 92). Then the work imparted per second is 

This becomes zero when u = o or when 11 = i\ and it is a 
maximum when u = \i\ or when the vane moves with one- 
half the velocity of the jet. Inserting this value of 21, 

k = (i + cos /i) IF — ; 



Art. 134.] WORK DERIVED FROM MOVING VANES. 325 

V 2 
and dividing this by the theoretic energy W — , the efficiency is 

e = -J(i -|- cos fi). 

When ft = 180°, the jet merely glides along the surface with- 
out performing work and e = o ; when /3 = 90 , the jet is 
deflected normally to the direction of the motion and e = % ; 
when /S = o°, a complete reversal of direction is obtained and 
the efficiency attains its maximum value e = :. 

Tfiese conclusions apply approximately to the vanes of a 
water-wheel which are so shaped that the water enters upon 
them tangentially in the direction of the motion. If the vanes 
are plane radial surfaces, as in simply paddle-wheels, the water 
'passes away normally to the circumference and the highest 
obtainable efficiency is about 50 per cent. If the vanes are 
curved backward the efficiency becomes greater, and, neglect- 
ing losses in impact and friction, it might be made nearly unity, 
and the entire energy of the stream be realized, if the water 
could both enter and leave the vanes in a direction tangent to 
the circumference. The investigation shows that this is due 
to the fact that the water leaves the vanes without velocity ; 
for, as the advantageous velocity of the vane is \v, the water 
upon its surface has the relative velocity v — \v = \v ; then, if 
p = o, as it leaves the vane its absolute velocity is \v — \v = o. 
If the velocity of the vanes is less or greater than half the 
velocity of the jet, the efficiency is lessened, although slight 
variations from the advantageous velocity do not practically 
influence the value of e. 

Prob. 160. A nozzle 0.125 feet in diameter, whose coeffi- 
cient of discharge is O.95, delivers water under a head of 82 
feet against a series of small vanes on a circumference whose 
diameter is 18.5 feet. Find the most advantageous velocity of 
revolution. 



3^6 DYNAMIC PRESSURE OF FLOWING WAFER. [Chap. XL 

Article 135. Revolving Vanes. 
When vanes are attached to an axis around which they 
move, as is the case in water wheels, the dynamic pressure 
which is effective in causing the motion is that tangential to 
the circumferences of revolution ; or at any given point this 
effective pressure is normal to a radius drawn from the point to 
the axis. In Fig. 94 are shown two cases of a rotating vane ; 
in the first the water passes outward or away from the axis, 
and in the second it passes inward or toward the axis. The 
reasoning, however, is general and will apply to both cases. 
At A, where the jet enters upon the vane, let v be its absolute 
velocity, V its velocity relative to the vane, and ti the velocity 
of the point A ; draw u normal to the radius r and construct 
the parallelogram of velocities as shown, a being the angle be- 
tween the directions of u and v, and <p that between u and V. 




At B, where the 
that point norm 
water relative to 
the resultant of 
departing water 
reverse direction 
tions of z\ and u 



Fig. 94 . 

water leaves the vane, let u x be the velocity of 
al to the radius r, , and V x the velocity of the 
the vane ; then constructing the parallelogram, 

u x and V % is v x , the absolute velocity of the 
Let fi be the angle between J\ and the 

of u x , and 6 be the angle between the direc- 



Art. 135.] REVOLVING VANES. 3 2 7 

The total dynamic pressure exerted in the direction of 
the motion will depend upon the values of the impulse in the 
entering and departing streams. The absolute impulse of the 

W 

water before entering is — v, and that of the water after leav- 

g 

W 
ing is — z>j. Let the components of these in the direction of 

the motion be designated by P and P x ; then, 

W W 

P = — v cos a. P. = — v. cos 6. 

g g 

These, however, cannot be subtracted to give the resultant 

dynamic pressure, as was done in the case of motion in a 

straight line, because their directions are not parallel, and the 

velocities of their points of application are not equal. The 

resultant dynamic pressure is not important in cases of this 

kind, but the above values will prove very useful in the next 

article in investigating the work that can be performed by the 

vane. 

The given data for a revolving vane are the angles a and /?, 
and the velocities v, 21, and u x . To find the auxiliary angle 6 the 
triangle at B between u x and l\ gives 

v 1 cos = u 1 — V x cos fi. 
When the motion is in a straight line the relative velocities V 
and V l are equal, if the friction is so slight that it can be 
neglected ; for a revolving vane, however, they are unequal, and 
the relation between them will be deduced in the next article. 

If n be the number of revolutions around the axis in one 
second, the velocities u and u x are 

u = 27irn, 11 j = 27i r x n, 

and accordingly the relation obtains, 

u r 



328 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI. 

or the velocities of the points of entrance and exit are directly 
proportional to their distances from the axis If r and r 1 are 
both infinity, u equals ti i and the case is that of motion in a 
straight line as discussed in Art. 133. 

Prob. 161. If a point 14 inches from the axis moves with a 
uniform velocity of 62 feet per second, how many revolutions 
does it make per minute? 

Art. 136. Work derived from Revolving Vanes. 

The investigation in Art. 134 on the work and efficiency of a 
revolving vane supposes that all its points move with the same 
velocity, and that the water enters upon it in the same direc- 
tion as that of its motion, or that a = o. This cannot in 
general be the case in water motors, as then the jet would be 
tangential to the circumference and no water could enter. To 
consider the subject further the reasoning of the last article 
will, be continued, and, using the same notation, it will be plain 
that the work may be regarded as that due to the impulse of 
the entering stream in the direction of the motion around the 
axis minus that due to the impulse of the departing stream in 
the same direction, or 

k = Pu - P x u v 

Here P and P 1 are the pressures due to the impulse at A and 
B (Fig. 94), and inserting their values as found, 

W 

k = — (uv cos a — u 1 v J cos 6) (S6) 

This is a general formula applicable to the work of all wheels 
of outward or inward flow, and it is seen that the useful work 
k consists of two parts, one due to the entering and the other 
to the departing stream. 

Another general expression for the work of a series of vanes 
may be established as follows : Let v and i\ be the absolute 



Art. 136.] WORK DERIVED FROM REVOLVING VANES. 329 

velocities of the entering and departing water ; then the theo- 
retic energy is W — •, and there is carried away the energy 

W — . The difference of these is the work imparted to the 

wheel, neglecting losses of energy in friction and impact, or 

W 
k = —(v*-v*) (87) 

This is a formula of equal generality with the preceding, and 
like it is applicable to all cases of the conversion of energy into 
work by means of impulse or reaction. In both formulas, how- 
ever, the plane of the vane is supposed to be horizontal, so 
that no fall occurs between the points of entrance and exit. 

A useful relation between the relative velocities V and V 1 
can be deduced by equating the values of k given by the pre- 
ceding formulas ; thus : 

mv cos a — u l v 1 cos 6 = -|(V — v x ). 

Now from the triangle at A between u and v 

v" = V' 2 — t? -f- 2uv cos a y 

and from the triangle at B between u x and v x 

7'j 2 = F/ — 71* -\- 2U 1 v 1 cos 0. 

Inserting these values of v 1 and v x 2 the relation reduces to 

V: - u; = V 2 - t? (88) 

This is the formula by which the relative velocity V x of the 
issuing' water is to be computed when V is given. It shows 
when u i = u that V x = V, as is the case in Fig. 93, where the 
motion is in a straight line. If, however, u x be greater than «, 
as in the outward-flow vane of the first diagram of Fig. 94, 
then V x is greater than V; if u x is less than u\ as in an inward- 
flow vane, then V x is less than V. 

The above principles will now be applied to the simple case t 



330 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XL 

of an ©utward-flow wheel driven by a fixed nozzle, as in the 
left-hand diagram of Fig. 95. The wheel is so built that 
r — 2 feet, ?\ = 3 feet, a = 45 , = 90 , and j3 = 30 . The 




Fig. 95. 

velocity of the water issuing from the nozzle is v = 100 feet per 
second and the discharge per second is 2.2 cubic feet. It is re- 
quired to find the work of the wheel and the efficiency when its 
speed is 337.5 revolutions per minute. 

The theoretic work of the stream per second is its weight 
multiplied by its velocity-head, or 



K = 



2.2 x 62.5 X ioo 2 



21377 foot-pounds, 



which gives 38.9 theoretic horse-powers. The actual work of the 
wheel, neglecting losses in foam and friction, can be computed 
either from (86) or (Sy). In order to use the first of these, how- 
ever, the velocities u, zt lt z\, and the angle 6 must be found, and 
to use the second z\ must be found; in either case Fand J\ 
must be determined. 

The velocities 71 and u x are found from the given speed of 
5.625 revolutions per second, thus : 

u = 2 X 3-1416 X 2 X 5-525 = 70.71 feet per second, 
«, = i| X 70-71 = 106.06 feet per second. 



Art. 136.] WORK DERIVED FROM REVOLVING VANES. 33 1 

The relative velocity Fat the point of entrance is found from 
the triangle between V and v, which in this case is right-angled, 

V ■=■ v cos (0 — a) = v cos 45 = 70.71 feet per second. 

The relative velocity V x at the point of exit is found from the 
relation (88), which gives V x = u x = 106.06 feet per second. 
And since u x and V x are equal, v x bisects the angle between V x 
and u x , and accordingly 

6 = i(i8o° - P) = 75 degrees. 

The value of the absolute velocity v x then is 

v x = 2u x cos = 54.90 feet per second, 

and v x /2g is the velocity-head lost in the escaping water. 

The work of the wheel per second, computed either from 
(86) or (87), is now found to be k — 14934 foot-pounds or 27.2 
horse-powers, and hence the efficiency, or the ratio of this work 
to the theoretic work, is e = 0.699. Thus 30.1 per cent of the 
energy of the water is lost, owing to the fact that the water 
leaves the wheel with such a large absolute velocity. 

In this example the speed given, 337.5 revolutions per min- 
ute, is such that the direction of the relative velocity V is 
tangent to the vane at the point of entrance. For any other 
speed this will not be the case, and thus work will be lost in 
shock and foam. It is observed also that the approach angle 
a is one-half of the entrance angle <p \ with this arrangement 
the velocities u and V are equal, as also u x and V x , and thus 
the absolute velocity of exit v x is made very small. These two 
conditions are essential in the construction of impulse wheels 
in order to secure high efficiency. 

Prob. 162. Compute the power and efficiency of the inward 
flow wheel in Fig. 95, when r = 3 feet, r x = 2 feet, a = 30 , 
= 6o°, (3 = 6o°, v = 100 feet per second, q = 2.2 cubic feet 
per second, and the speed being 184 revolutions per minute. 



33 2 DYNAMIC PRESSURE OF FLOWING WATER. [Chap. XI. 



Article 137. Revolving Tubes. 

The water which glides over a vane can never be under 
static pressure, but when two vanes are placed near together 
and connected so as to form a closed tube, there may exist in 
it static pressure if the tube is filled. This is the condition in 
turbine wheels, where a number of such tubes, or buckets, are 
placed around an axis and water is forced through them by 
the static pressure of a head. The work in this case is done 
by the dynamic pressure exactly as in vanes, but the existence 
of the static pressure renders the investigation more difficult. 

The simplest instance of a revolving tube is that of an arm 
attached to a vessel rotating about a vertical 
axis, as in Fig. 96. It was shown in Art. 29 
that the water surface in this case assumes 
the form of a paraboloid, and if no discharge 
occurs it is clear that the static pressures at 
any two points B and A are measured by 
the pressure-heads //", and H reckoned up- 
wards to the parabolic curve, and, if the ve- 
locities of those points are u y and u, that 





H, 



? c 


7f 


— =H- 




*g 


lg 



h. 



Now suppose an orifice to be opened in the 
FlG ' g6 ' end of the tube and the flow to occur while 

at the same time the revolution is continued. The velocities 
V 1 and V diminish the pressure-heads so that the piezometric 
line is no longer the parabola but some curve represented by the 
lower broken line in the figure. Then according to the prin- 
ciple in Art. 27, that pressure-head plus velocity-head remains 
constant if no loss of energy occurs, the above equation be- 
comes 



#> + 



V: 



H -V 



V 1 



(39) 



Art. I37-] REVOLVING TUBES. 333 

in which H l and H are the heads due to the actual static pres- 
sures. This is the theorem which gives the relation between 
pressure-head, velocity-head, and rotation-head at any point 
of a revolving tube or bucket. If the tube is only partly full, 
so that the flow occurs along one side, like that of a stream 
upon a vane, then there is no static pressure, U 1 = o, H = o^ , 
and the formula becomes the same as (88), which was otherwise 
deduced in the last article. 

An apparatus like Fig. 96, but having a number of arms 
from which the flow issues, is called a reaction wheel, since the 
dynamic pressure which causes the revolution is wholly due to 
the reaction of the issuing water. To investigate it, the gen- 
eral formula (86) may be used. Making u = o, then the work 
done upon the wheel by the water is 

W W 

k = — (— u x v x cos 6) = — {u x V x cos (3 — u*). 

But since there is no static pressure at the point B, the value 
of V x is, from (89), or also from Art. 29, 



V x = V2gh + u*. 
The work of the wheel now is 
W 



k — — {tt l cos /? V2gh -f- u? — u*). 

This becomes nothing when u x = o, or when n x = 2gh cot 2 J3, 
and by the usual method it is found that it becomes a maxi- 
mum when 

< = -£-3 - gh . 
sin (3 6 

Inserting this advantageous velocity, the corresponding work is 

k = Wh{\ - sin (3), 
and therefore the efficiency is 

e = 1 — sin (3. 



334 DYNAMIC PRESSURE OF FLOWING WATER. [Chap XL 

When ft = 90 , both «, and e become o, for then the direction 
of the stream is normal to the circumference of revolution and 
no reaction can occur. When ft = o the efficiency becomes 
unity, but the velocity u x becomes infinity. In the reaction 
wheel, therefore, high efficiency can only be secured by making 
the direction of the issuing water directly opposite to that 
of the revolution, and by having the speed very great. If 
ft = 1 9 . 5 or sin ft = -J-, the advantageous velocity u 1 becomes 
Vigh and e becomes 0.67. The effect of friction of the water 
on the sides of the revolving tube is not here considered, but 
in Art. 154, where the reaction wheel is to be further discussed^ 
this will be done. 

Prob. 163. Compute the theoretic efficiency of the reaction 
wheel when 6 = 180 , ft = o°, and u x = V2gh. 



Art. 138.] GENERAL PRINCIPLES. 335 



CHAPTER XII. 
NAVAL HYDROMECHANICS. 

Article 138. General Principles.. 

In this chapter is to be discussed in a brief and elementary 
manner the subject of the resistance of water to the motion of 
vessels, and the general hydrodynamic principles relating to 
their propulsion. The water may be at rest and the vessel 111 
motion — or both may be in motion, as in the case of a boat go- 
ing up or down a river. In either event the velocity of the 
vessel relative to the water need only be considered, and this 
will be called v. The simplest method of propulsion is by the 
oar or paddle ; then come the paddle wheel, and the jet and 
screw propellers. The action of the wind upon sails will not 
be here discussed, as lying outside of the scope of the work. 

The unit of measure used on the ocean is generally the nau- 
tical mile or knot, which is about 6080 feet, so that knots per 
hour may be transformed into feet per second by multiplying 
by 1.69, and feet per second may be transformed into knots per 
hour by multiplying by 0.592. On rivers the statute mile is 
used, and the corresponding multipliers will be 1.47 and 0,682. 
On the ocean the weight of a cubic foot of water is to be taken 
as about 64 pounds (it is often used as 64.32 pounds, so that 
the numerical value is the same as 2g), and in rivers at 62.5 
pounds. 

The speed of a ship at sea is roughly measured by observa- 
tions with the log, which is a triangular piece of wood attached 
to a cord which is divided by tags into lengths of about 5 of 



3j6 NAVAL HYDROMECHAXICS. [Chap. XII. 

feet. The log being thrown, the number of tags run out in half 
a minute is the same as the number of knots per hour at which 
the ship is moving, since 5 of feet is the same part of a knot that 
a half minute is of an hour. The patent log, which is a small 
self-recording current meter, drawn in the water behind the 
ship, is however now generally used. In experimental work 
more accurate methods of measuring the velocity are necessary, 
and for this purpose the boat may run between buoys whose 
distance apart has been found by triangulation from measured 
bases on shore. 

When a boat or ship is to be propelled through water, the 
resistances to be overcome increase with its velocity, and con- 
sequently, as in railroad trains, a practical limit of speed is soon 
attained. These resistances consist of three kinds — the dynamic 
pressure caused by the relative velocity of the boat and the 
water, the frictional resistance of the surface of the boat, and 
the wave resistance. The first of these can be entirely over- 
come, as indicated in Art. 132, by giving to the boat a " fair'* 
form, that is, such a form that the dynamic pressure of the im- 
pulse near the bow is balanced by that of the reaction of the 
water as it closes in around the stern. It will be supposed in 
the following pages that the boat has this form, and hence this 
first resistance need not be further considered. The second 
and third sources of resistance will be discussed later. 

The total force of resistance which exists when a vessel is 
propelled with the velocity v can be ascertained by drawing it 
in tow at the same velocity, and placing on the tow line a dy- 
namometer to register the tension. An experiment by FROUDE 
on the Greyhound, a steamer of 11 57 tons, gave for the total 
resistance the following figures : * 

At 4 knots per hour, 0.6 tons ; 

At 6 knots per hour, 1.4 tons ; 

* Thearle's Theoretical Naval Architecture. London. 1S76, p. 347. 



Art. 139.] FRICTIONAL RESISTANCES. 337 

At 8 knots per hour, 2.5 tons; 
At 10 knots per hour, 4.7 tons ; 
At 12 knots per hour, 9.0 tons. 

This shows that at low speeds the resistance varies about as 
the square of the velocity, and at higher speeds in a faster 
ratio. For speeds of 15 to 18 knots per hour— the usual ve- 
locity of ocean steamers — there is but little known regarding 
the resistance, but as an approximation it is usually taken as 
varying with the square of the velocity. 

Prob. 164. What horse-power was expended in the above 
test of the Greyhound when the speed was 12 knots per hour? 

Article 139. Frictional Resistances. 

When a stream or jet moves over a surface its velocity is 
retarded by the frictional resistances, or if the velocity be main- 
tained uniform a constant force is overcome. In pipes, con- 
duits, and channels of uniform section the velocity is uniform, 
and consequently each square foot of the surface or bed exerts 
a constant resisting force, the intensity of which will now be 
approximately computed. This resistance will be the same as 
the force required to move the same surface in still water, and 
hence the results will be directly applicable to the propulsion 
of ships. 

Let F be the force of frictional resistance per square foot 
of surface of the bed of a channel, p its wetted perimeter, / its 
length, h its fall in that length, a the area of its cross-section, 
and v the mean velocity of flow. The force of friction over 
the entire surface then is Fpl, and the work per second lost in 
friction is Fplv. The work done by the water per second : s 
Wh or wavh. Equating these two expressions for the work, 

ah 

Jp — w ~- 7 = wrs. 
pi 



33% NA VAL HYDROMECHANICS. [Chap. XII. 

Now, inserting for rs its value from formula (70) of Art. 94, 
there results 

c 

in which w is the weight of a cubic foot of water, and c is the 
coefficient in the mean velocity formulas whose value is to be 
taken from the tables in Chapter VIII. Inasmuch as the ve- 
locities along the bed of a channel are somewhat less than the 
mean velocity v, the values of F thus determined will probably 
be slightly greater than the actual resistance. 

For smooth iron pipes the following are values of the fric- 
tional resistance in pounds per square foot of surface at differ- 
ent velocities, as computed from the above formula : 





V = 2. 


4- 


6. 


10. 


15- 


For 1 foot diameter, 


F — 0.023, 


O.080, 


O.I7, 


O.43, 


0.92 ; 


For 4 feet diameter, 


F = 0.015, 


O.053, 


0. 1 1 , 


0.28, 


0.59. 



These figures indicate that the resistance is subject to much 
variation in pipes of different diameters ; it is not easy to con- 
clude from them, or from formula (70), what the force of re- 
sistance is for plane surfaces over which water is moving. 

Experiments made by moving flat plates in still water so 
that the direction of motion coincides with the plane of the 
surface have furnished conclusions regarding the laws of fluid 
friction similar to those deduced from the flow of water in pipes. 
It is found that the total resistance is approximately propor- 
tional to the area of the surface, and approximately propor- 
tional to the square of the velocity. Accordingly, the force of 
resistance per square foot may be written 

F = fv\ 

in which / is a number depending upon the nature of the sur- 



Art. 139.] FRICTIONAL RESISTANCES. 339 

face. The following are average values of / for large surfaces, 
as given by Unwin:* 

Varnished surface, f == 0.00250; 

Painted and planed plank, f — 0.00339; 

Surface of iron ships, f = 0.00351 ; 

Fine sand surface, f = 0.00405 ; 

New well-painted iron plate, f = 0.00473. 

Undoubtedly the value of f is subject to variations with the 
velocity, but the experiments on record are so few that the law 
and extent of its variation cannot be formulated. It should, 
however, be remarked that the formulas and constants here 
given do not apply to low velocities, for the reasons given in 
Art. 92. At the same time they are only approximately ap- 
plicable to high velocities A low velocity of a body moving 
in an unlimited stream may be regarded as 1 foot per second 
or less, a high velocity as 25 or 30 feet per second. 

It may be noted that the above-mentioned experiments in- 
dicate that the value of F is greater for small surfaces than for 
large ones. For instance, a varnished board 50 feet long gave 
f = 0.00250, while one 20 feet long gave f = 0.00278, and one 
8 feet long gave f = 0.00325, the motion being in all cases in 
the direction of the length. The resistance is the same what- 
ever be the depth of immersion, for the friction is uninfluenced 
by the intensity of the static pressure. This is proved by the 
circumstance that the flow of water in a pipe is found to de- 
pend only upon the head on the outlet end, and not upon the 
pressure-heads along its length. 

Prob. 165. What is the frictional resistance of a boat when 
moving at the rate of 9 knots per hour, the area of its immersed 
surface being 320 square feet, and f — 0.0035 ? 

* Encyclopaedia Britannica, 9th Edition, vol. xii. p. 483. 



540 NAVAL HYDROMECHANICS. [Chap. XI I.. 

Article 140. Work required in Propulsion. 

When a boat or ship moves through still water with a ve- 
locity v, it must overcome the pressure due to impulse of the 
water and the resistance due to the friction of its surface on the 
water and air. If the surface be properly curved there is no 
resultant pressure due to impulse, as shown in Art. 132. The 
resistance caused by friction of the immersed surface on the 
water can be estimated, as explained above. If A be the area 
of this surface in square feet, the work per second required to 
overcome this resistance is 

k = AFv =fAv\ 

The work, and hence the horse-power, required to move a boat 
accordingly varies approximately as the cube of its velocity. 
By the help of the values of /given in the last article an ap- 
proximate estimate of the work can be made for particular 
cases. The resistance of the air, which in practice must be 
considered, will be here neglected. 

To illustrate this law let it be required to find how many 
tons of coal will be used by a steamer in making a trip of 3000 
miles in 6 days, when it is known that 800 tons are used in 
making the trip in 10 days. As the power used is proportional 
to the amount of coal, and as the distances travelled per day in 
the two cases are 500 miles and 300 miles, the law gives 

T 5_ 3 

6X80 3 3 ' 

whence T = 2220 tons. By the increased speed the expense 
for fuel is increased 277 per cent, while the time is reduced 40 
per cent. If the value of wages, maintenance, interest, etc., 
saved on account of the reduction in time, will balance the 
extra expense for fuel, the increased speed is profitable. That 
such a compensation occurs in many instances is apparent from 



Art. 141.] THE JET PROPELLER. 34 1 

the constant efforts to reduce the time of trips of passenger 
steamers. 

When a boat moves with the velocity v in a current which 
has a velocity u in the same direction the velocity of the boat 
relative to the water is v — 21, and the resistance is proportional 
to (v — it) 2 and the work to (y — uf. If the boat moves in the 
opposite direction to the current the relative velocity is v -\- u, 
and of course v must be greater than u or no progress would 
be made. In all cases of the application of the formulas of 
this article and the last, v is to be taken as the velocity of the 
boat relative to the water. 

Another source of resistance to the motion of boats and 
ships is the production of waves. This is due in part to a 
different level of the water surface along the sides of the ship 
due to the variation in static pressure caused by the velocity, 
and in part to other causes. It is plain that waves, eddies, and 
foam cause energy to be dissipated in heat, and that thus a 
portion of the work furnished by the engines of the boat is lost. 
This source of loss is supposed to consume from 10 to 40 per 
cent of the total work, and it is known to increase with the 
velocity. On account of the uncertainty regarding this resist- 
ance, as well as those due to the friction of the water and air, 
practical computations on the power required to move boats at 
given velocities can only be expected to furnish approximate 
results. 

Prob. 166. Compute the horse-power required for a ve- 
locity of 18 knots per hour, taking A = 7473 square feet and 
/ = 0.004. 

Article 141. The Jet Propeller. 

The method of jet propulsion consists in allowing water to 
enter the boat and acquire its velocity, and then to eject it 
backwards at the stern by means of a pump. The reaction thus 



342 NAVAL HYDROMECHANICS. [Chap. XI L 

produced propels the boat forward. To investigate the effi- 
ciency of this method, let W be the weight of water ejected 
per second, V its velocity relative to the boat, and v the ve- 
locity of the boat itself. The absolute velocity of the issuing 
water is then V — v, and it is plain without further discussion 
that the maximum efficiency will be obtained when this is o, 
or when V = v, as then there will be no energy remaining 
in the water which is propelled backward. It is, however, to 
be shown that this condition can never be realized. 

The work which is lost in the absolute velocity of the 

water is 

W 

k' = — (V- v)\ 

2g 

The work wmich is exerted on the boat by the reaction is 

W 

k= — (V- v)v. 

g 

The sum of these is the total theoretic work, or 

W 
K=—(V*- zr). 
2g } 

Therefore the efficiency of jet propulsion is expressed by 

k 2V 






K V+v' 

This becomes equal to unity when v = F as before indicated, 
but then it is seen that the work k becomes o unless IF is infinite. 
The value of Wis waV, if a be the area of the orifices through 
which the water is ejected ; and hence in order to make e unity 
and at the same time perform work it is necessary that either 
V ox a should be infinity. The jet propeller is therefore like 
a reaction wheel (Art. 152), and it is seen upon comparison that 
the formula for efficiency is the same in the two cases. 

By equating the above value of the useful work to that 
established in the last article there is found 
fgAv % = waV{V- v)\ 



Art. 142.] PADDLE WHEELS. 343 

and if this be solved for V, and the resulting value be substi- 
tuted in (100), it reduces to 

4 






V" 



3 + V 1 + 



'which again shows that e approaches unity as the ratio of a to 
A increases. The area of the orifices of discharge must hence 
be very large in order to realize both high power and high 
efficiency. For this reason attempts to propel vessels by this 
method have not proved practically successful. In nature the 
same result is seen, for no marine animal except the cuttle-fish 
uses this principle of propulsion. Even the cuttle-fish cannot 
depend upon his jet to escape from his enemies, but for this 
relies upon his supply of ink, with which he darkens the water 
about him. 

Prob. 167. Compute the velocity and efficiency of a jet pro- 
peller driven by a i-inch nozzle under a pressure of 150 pounds 
per square inch, when A = 1000 square feet and/"— 0.004. 

Article 142. Paddle Wheels. 

The method of propulsion by rowing and paddling is famil- 
iar to all. The power is furnished by muscular energy within 
the boat, the water is the fulcrum upon which the blade of the 
oar acts, and the force of reaction thus produced is transmitted 
to the boat and urges it forward. If water were an unyielding 
substance, the theoretic efficiency of the oar should be unity, 
or, as in any lever, the work done by the force at the rowlock 
should equal the work performed by the motive force exerted* 
But as the water is yielding, some of it is driven backward by 
the blade of the oar, and thus energy is lost. 

The paddle or side wheel so extensively used in river navi- 
gation is similar in principle to the oar. The former is furnished 



344 NA VAL HYDROMECHANICS. [Chap. XII. 

by a motor within the boat, the blades or vanes of the wheel 
tend to drive the water backward, and the reaction thus pro- 
duced urges the boat forward. On first thought it might be 
supposed that the efficiency of the method would be governed 
bylaws similar to those of the undershot wheel, and such would 
be the case if the vessel were stationary and the wheel were 
used as an apparatus for moving the water. In fact, however, 
the theoretic efficiency of the paddle wheel is much higher than 
that of the undershot motor. 

The work exerted by the steam-engine upon the paddle 
wheels may be represented by PV, in which P is the pressure 
produced by the vanes upon the water, and V is their velocity 
of revolution ; and the work actually imparted to the boat may 
be represented by Pv, in which v is its velocity. Accordingly 
the efficiency of the paddle, neglecting losses due to foam and 

waves, is 

v v 

e = 



V v + vl 

in which v 1 is the difference V — v, or the so-called " slip." If 
the slip be o, the velocities Fand v are equal, and the theoretic 
efficiency is unity. The value of V is determined from the 
radius r of the wheel and its number of revolutions per minute ; 
thus V= 2nrN. 

On account of the lack of experimental data it is difficult to 
give information regarding the practical efficiency of paddle 
wheels considered from a hydromechanic point of view. Ow- 
ing to the water which is lifted by the blades, and to the foam 
and waves produced, much energy is lost. They are, however, 
very advantageous on account of the readiness with which the 
boat can be stopped and reversed. When the wheels are driven 
by separate engines, as is sometimes done on river boats, per- 
fect control is secured, as they can be revolved in opposite 
directions when desired. Paddle wheels with feathering blades 



Art. 143.] THE SCREW PROPELLER. 345 

are more efficient than those with fixed radial ones, but prac- 
tically they are found to be cumbersome, and liable to get out 
of order.* In ocean navigation the screw has now almost en- 
tirely replaced the paddle wheel on account of its higher 
efficiency. 

Prob. 168. Ascertain the size of the paddle wheels of the 
steamship Great Eastern. 

Article 143. The Screw Propeller. 

The screw propeller consists of several helicoidal blades 
attached at the stern of a vessel to the end of a horizontal 
shaft which is made to revolve by steam power. The dynamic 
pressure of the reaction developed between the water and the 
helicoidal surface drives the vessel forward, the theoretic work 
of the screw being the product of this pressure by the distance 
traversed. The pitch of the screw is the distance, parallel to 
the shaft, between any point on a helix, and the corresponding- 
point on the same helix after one turn around the axis, and 
the pitch may be constant at all distances from the axis, or it 
may be variable. If the water were unyielding, the vessel 
would advance a distance equal to the pitch at each revolution 
of the shaft ; actually, the advance is less than the pitch, the 
difference being called the slip. The effect thus is that the 
pressure P existing between the helical surfaces and the water 
moves the vessel with the velocity v, while the theoretic velocity 
which should occur is V, being the pitch of the screw multi- 
plied by the number of revolutions per second. The work 
expended is hence PV or P(v -\- ?>,), if z\ be the slip per second, 
and the work utilized isPv. Accordingly the efficiency of screw 
propulsion is, approximately, 



v-\- v' 



For description of these, see Knight's Mechanical Dictionary. 



346 NAVAL HYDROMECHANICS. [Chap. .XII. 

which is the same expression as before found for the paddle 
wheel. Here, as in the last article, all the pressure exerted 
by the blades upon the water is supposed to act backward in a 
direction parallel to the shaft of the screw, and the above 
conclusion is approximate because this is actually not the case,, 
and also because the action of friction has not been considered. 

The pressure P which is exerted by the helicoidal blades 
upon the water is the same as the thrust or stress in the shaft, 
and the value of this may be approximately ascertained by re- 
garding it as due to the reaction of a stream of water of cross- 
section a and velocity v, or 

/»=f (*.+•> 

Another expression for this may be found from the expended 
work k ; thus : 

v 

Numerical values computed from these two expressions do 
not, however, agree well, the latter giving in general a much 
less value than the former. 

In Art. 149 the work to be performed in propelling a vessel 
of fair form whose submerged surface is A was found to be 

k=fAv\ 

If the value of v is taken from this and inserted in the ex- 
pression for efficiency, there obtains 

1 



e = 



1* 



which shows that e increases as v i , f, and A decrease, and as 
/'increases. Or for given values of /and A the efficiency de- 
creases with the speed. 



Art. 143.] 



THE SCREW PROPELLER. 



347 



It has been observed in a few instances that the slip v 1 is 
negative, or that V, as computed from the number of revolu- 
tions and pitch of the screw, is less than v. This is probably 
due to the circumstance that the water around the stern is 
following the vessel with a velocity v' f so that the real slip is 
V — v -\- v' instead of V — v. The existence of negative slip 
is usually regarded as evidence of poor design. 

In some cases twin screws are used, as with these the vessel 
can be more readily controlled. Fig. 97 shows the twin 
screws of the City of New York, an ocean steamer of 580 feet 




Fig 97- 

length, 63.5 feet breadth, and 42 feet depth, with a gross ton- 
nage of 10 500 and an estimated horse-power of about 1600a 
These are made to revolve in opposite directions. The usual 
practice, however, is to have but one screw. The practical 
advantage of the screw over the paddle wheel has been found 
to be very great, and this is probably due to the circumstance 
that less energy is wasted in lifting the water and in forming 
waves. 



Prob. 169. Ascertain the size and the pitch of the screws 
on the steamer City of New York. Compute the theoretic 
efficiency if the number of revolutions per minute is 150 when 
the velocity of the steamer is 20 knots per hour. 



343 



NA VAL HYDROMECHAXICS. 



[Chap. XII. 



Article 144. The Action of the Rudder. 

The action of the rudder in steering a vessel involves a 
principle that deserves discussion. In Fig. 108 is shown a plan 

of a boat with the rudder turned to 
the starboard side, at an angle 6 
with the line of the keel. The ve- 
locity of the vessel being v, the ac- 
tion of the water upon the rudder is 
the same as if the vessel were at rest 
Fig. 98. and the water in motion with the 

velocity v. Let Wbe the weight of water which produces dy- 

W 

namic pressure against the rudder, due to the impulse — v (Art. 

128). The component of this pressure normal to the rudder is 

W 




P = 



sin 



and its effect in turning the vessel about the centre of gravity 
C is measured by its moment with reference to that point. 
Let b be the breadth of the rudder, and d the distance CH be- 
tween the centre of gravity and the hinge of the rudder ; then 
the lever arm of the force P is 

/= \b + d qos 6, 

and accordingly the turning moment is 

W 

M = —v{b sin 6 + d sin 20). 

To determine that value of which produces the greatest effect 
in turning the boat the derivative of M with respect to must 
vanish, which saves 



b /i lr 



Art. 145.J TIDES AND WAVES. 349 

and from this the value of is found to be approximately 45 , 
since d is always much larger than b. 

The following are values of 6 for several values of the 
ratio b -=- d: 

b^d— jr 1- T \ ^ o 

cos 6 = 0.6825 0.6916 0.6947 0.7069 0.7071 
# = 4 6°58 / 46°i5 / 46°oo / 45°oi' 45 

In practice it is usual to arrange the mechanism of the rudder 
so that it can only be turned to an angle of about 42 with the 
keel, for it is found that the power required to turn it the addi- 
tional 3 or 4 is not sufficiently compensated by the slightly 
greater moment that would be produced. The reasoning also 
shows that intensity of the turning moment increases with v, so 
that the rudder acts most promptly when the boat is moving 
rapidly. For the same reason a rudder on a steamer propelled 
by a screw does not need to be so broad as on one driven by 
paddle wheels, for the effect of the screw is to increase the ve- 
locity of the impinging water, and hence also its dynamic pres- 
sure against the rudder. 

Prob. 170. Explain how it is that a ship can sail against the 
wind. 

Article 145. Tides and Waves. 

The complete discussion of the subject of waves might, like 
so many other branches of Hydraulics, be expanded so as to 
embrace an entire treatise, and hence there can be here given 
only the briefest outline of a few of the most important prin- 
ciples. There are two classes or kinds of waves, the first in- 
cluding the tidal waves and those produced by earthquakes or 
other sudden disturbances, and the second those due to the 
wind. The daily tidal wave generated by the attraction of the 
moon and sun originates in the South Pacific Ocean, whence 



35o 



. NAVAL HYDROMECHANICS. 



[Chap. XII. 



it travels in all directions with a velocity dependent upon the 
depth of water and the configuration of the continents, and 
which in some regions is as high as iooo miles per hour. 
Striking against the coasts, the tidal waves cause currents in 
inlets and harbors, and if the circumstances were such that 
their motion could become uniform and permanent, these might 
be governed by the same laws which apply to the flow of water 
in channels. Such, however, is rarely the case ; and according- 
ly the subject of tidal currents is one of much complexity, and 
not capable of general formulation. 

The velocity of a wave produced by a sudden disturbance 
in a channel of uniform width is found by experiments, and 
also by theoretic considerations, to be 4 gD, where D desig- 
nates the depth of water. When such a wave advances into 
shallow water its height is observed to increase, and when D 
becomes as small as one-half the height of the wave it breaks 
into foam. 

Rolling waves produced by the wind travel with a velocity 
which is small compared with those of the first class, although 
in water where the disturbance can extend to the bottom it is 
generally supposed that their speed is also represented by 




i gD. Upon the ocean the maximum length of such waves is 
estimated at 550 feet, and their velocity at about 53 feet per 
second. For this class of waves it is found by observation that 
each particle of water upon the surface moves in an elliptic or 
circular orbit, whose time of revolution is the same as the time 



Art. 1 45-] TIDES AND WAVES. 35 l 

of one wave length. Thus the particles on the crest of a wave 
are moving forward in the direction of the motion of the wave, 
while those in the trough are moving backward. When such 
waves advance, into shallow water their length and speed de- 
crease, but the time of revolution of the particles in their orbits 
remains unaltered, and as a consequence the slopes become 
steeper and the height greater, until finally the front slope be- 
comes vertical, and the wave breaks with roar and foam. Be- 
low the surface the particles revolve also in elliptic orbits, which 
grow smaller in size toward the bottom. The curve formed 
by the vertical section of the surface of a wave at right angles 
to its length is of a cycloidal nature. 

The force exerted by ocean waves when breaking against 
sea walls is very great, as already mentioned in Art. 132, and 
often proves destructive. If walls can be built so that the 
waves are reflected without breaking, as is sometimes possible 
in deep water, their action is rendered less injurious. Upon 
the ocean waves move in the same direction as the wind, but 
along shore it is observed that they move normally toward it, 
whatever may be the direction in which the wind is blowing. 

Prob. 171. In a channel 6.5 feet wide, and of a depth de- 
creasing uniformly 1.5 feet per 1000, Bazin generated a wave 
by suddenly admitting water at the upper end. At points 
where the depths were 2.16, 1.85, 1.46, and 0.80 feet, the ve- 
locities were observed to be 8.70, 8.67, 7.80, and 6.69 feet per 
second. Do these velocities agree with the law above stated ? 



35 2 WATER WHEELS. [Chap. XII L 



CHAPTER XIII. 
WATER WHEELS. 

Article 146. Conditions of High Efficiency 

A hydraulic motor is an apparatus for utilizing the energy 
of a waterfall. It generally consists of a wheel which is 
caused to revolve, either by the weight of water falling from a 
higher to a lower level, or by the dynamic pressure due to the 
change in direction and velocity of a moving stream. When 
the water enters at only one part of the circumference the 
apparatus is called a water wheel ; when it enters around the 
entire circumference it is called a turbine. In this chapter 
and the next these two classes of motors will be discussed in 
order to determine the conditions which render them most 
efficient. 

The efficiency e of a motor ought, if possible, to be inde- 
pendent of the amount of water used, or if not, it should be the 
greatest when the water supply is low. This is very difficult to 
attain. It should be noted, however, that it is not the mere 
variation in the quantity of water which causes the efficiency 
to vary, but it is the losses of head which are consequent 
thereon. For instance, when water is low, gates must be low- 
ered to diminish the area of orifices, and this produces sudden 
changes of section which diminish the effective head h. A 
complete theoretic expression for the efficiency will hence not 
include W, the weight of water supplied per second, but it 
should, if possible, include the losses of energy or head which 
result when W varies. The actual efficiency of a motor can 



Art. 146.] CONDITIONS OF HIGH EFFICIENCY. . 353 

only be determined by tests with a friction brake ; the theoretic 
efficiency, as deduced from formulas like those of Chapter XI, 
will as a rule be higher than the actual, because it is impossible 
to formulate accurately all the sources of loss. Nevertheless, 
the deduction and discussion of formulas for theoretic efficiency 
is very important for the correct understanding and successful 
construction of hydraulic motors. 

The theoretic energy per second of W pounds of water fall- 
ing through a height of h feet, or moving with a velocity v, is 

K= Wk= W-. 

The actual work per second equals the theoretic energy minus 
all the losses of energy. These losses may be divided into two 
classes : first, those caused by the transformation of energy into 
heat; and second, those due to the velocity z\ with which the 
water reaches the level of the tail race. The first class includes 
losses in friction, losses in foam and eddies consequent upon 
sudden changes in cross-section, or from allowing the entering 
water to dash improperly against surfaces ; let the loss of work 
due to this be Wh f , in which h! is the head lost by these causes. 
The second loss is due merely to the fact that the departing 

water carries away the energy W ' — . The work per second 

imparted to the wheel then is 

and dividing this by the theoretic energy, the efficiency is 

-:-£-&)■ w 

This formula, although very general, must be the basis of all 
discussions on the theory of water-wheels and motors. It 
shows that e can only become unity when h' = o and v 1 = o, 



354 WATER WHEELS. [Chap. XIII. 

whence the two following fundamental requirements must be 
fulfilled in order to secure high efficiency : 

i. The water must enter and pass through the wheel with- 
out losing energy in friction and foam. 

2. The water must reach the level of the tail race without 
absolute velocity. 

These two requirements are expressed in popular language by 
the maxim, well known among engineers, "the water must 
enter the wheel without shock and leave without velocity." 
Here the word shock means that method of introducing the 
water which produces foam and eddies. 

The friction of the wheel upon its bearings is included in 
the lost work when the power and efficiency are actually 
measured as described in Art. 124. But as this is not a 
hydraulic loss, it should not be included in the lost work k' 
when discussing the wheel merely as a user of water, as will be 
done in this chapter. The amount lost in shaft and journal 
friction in good constructions may be estimated at 2 or 3 per 
cent of the theoretic energy, so that in discussing the hydraulic 
losses the maximum value of e will not be unity, but about 
0.98 or 0.97. This may perhaps be rendered slightly smaller 
by the friction of the wheel upon the air or water in which it 
moves, and which will here not be regarded. 

Prob. 172. A wheel using 70 cubic feet per minute under a 
head of 12.4 feet has an efficiency of 0.63. What is its effective 
horse-power? 

Article 147. Overshot Wheels. 

In the overshot wheel the water acts largely by its weight. 
Fig. 100 shows an end view or vertical section, which so fully 
illustrates its action that no detailed explanation is necessary. 
The total fall from the surface of the water in the head race or 
flume to the surface in the tail race is called //. The weight of 



Art. 147.] 



O VER SHO T I VHEEL S. 



355 



Avater delivered per second is represented by W\ then the 
theoretic energy of the fall per second is W/i. It is required 
to determine the conditions which will render the work of the 
wheel as near to Wh as possible. 




Fig. 100. 

The total fall may be divided into three parts — that in which 
the water is filling the buckets, that in which the water is 
descending in the filled buckets, and that which remains after 
the buckets are emptied. Let the first of these parts be called 
k , and the last h x . In falling the distance /i the water acquires 
a velocity v n which is approximately equal to V2gh , and then 
striking the buckets this is reduced to u, the tangential velocity 
of the wheel, whereby a loss of energy in impact occurs. It 
then descends through the distance h — k — h x acting by its 
weight alone, and finally dropping out of the buckets, reaches 
the level of the tail race with a velocity which causes a second 
loss of energy. Let h! be the head lost in entering the buckets, 
and let v 1 be the velocity of the water as it reaches the tail 
race. Then the efficiency of the wheel is given by the general 
formula (90), or 



e — 1 



77 



356 WATER WHEELS. [Chap. XIII. 

and to apply it, the values of h! and v 1 are to be found. In this 
equation v is the velocity due to the head h, or v = V2gh. 

The head lost when a stream of water with the velocity v 9 
is enlarged in section so as to have the smaller velocity u, is, as 
proved in Art. 68, 

k , = fa - ")* = T' 2 - 2z\u + u* 
2g tg 

The velocity v 1 with which the water reaches the tail race de- 
pends upon the velocity u and the height h l . Its energy as it 

leaves the buckets is W — , and that required in the fall h x is 

Wh y ; the sum of these must be equal to the resultant energy, 

,_, 2 

W '— , whence the value of v, is 

v x — Vu* -\- 2gh x . 

Inserting these values of li and z\ in the formula for e, and 
placing for v 7 its equivalent 2g/i, there is found 

r 2 - 2v a u + 2?/ 2 + 2gh, 

e = I — ; . 

2gh 

The value of u which renders e a maximum is by the usual 
method found to be 

u = iv, ; 

or the velocity of the wheel should be one-half that of the eru 
tering water. Inserting this value, the efficiency corresponding 
to the advantageous velocity is 



2srh 



and lastly, replacing v* by its value 2gh Q , it becomes 

i /t h x 



Art. 147.] OVERSHOT WHEELS. 357 

which is the theoretic maximum efficiency of the overshot 
wheel. 

This investigation shows that one-half of the entrance fall 
h a and the whole of the exit fall h x are lost, and it is hence plain 
that in order to make e as large as possible both /z and h x 
should be as small as possible. The fall /i is made small by 
making the radius of the wheel large ; but it cannot be zero, for 
then no water would enter the wheel : it is generally taken so 
as to make the angle about 10 or 15 degrees. The fall h x is 
made small by giving to the buckets a form which will retain 
the water as long as possible. As the water really leaves the 
wheel at several points along the lower circumference, the value 
of h x cannot usually be determined with exactness. 

The practical advantageous velocity of the overshot wheel, 
as determined by the method of Art. 124, is found to be about 
o.4^ , and its efficiency is found to be high, ranging from 70 to 
90 per cent. In times of drought, when the water supply is 
low, and it is desirable to utilize all the power available, its effi- 
ciency is the highest, since then the buckets are but partly 
filled and h x becomes small. Herein lies the great advantage 
of the overshot wheel; its disadvantage is in its large size and 
the expense of construction and maintenance. 

The number of buckets and their depth are governed by no 
laws except those of experience. Usually the number of buck- 
ets is about $r or 6r, if r is the radius of the wheel in feet, and 
their radial depth is from 10 to 15 inches. The breadth of the 
wheel parallel to its axis depends upon the quantity of water 
supplied, and should be so great that the buckets are not fully 
filled with water, in order that they may retain it as long as 
possible and thus make h 1 small. The wheel should be set with 
its outer circumference at the level of the tail water. 

Prob. 173. Estimate the horse-power of an overshot wheel 
which uses 1080 cubic feet of water per minute under a head 



53 



WATER WHEELS. 



[Chap. XIIL 



of 26 feet, the diameter of the wheel being 23 feet, and the 
water entering at 15 from the top and leaving at 12 from the 
bottom. 



Article 148. Breast Wheels, 

The breast wheel is applicable to small falls, and the action 
of the water is partly by impulse and partly by weight. As 
represented in Fig. 10 1, water from a reservoir is admitted 
through an orifice upon the wheel under the head /i with the 
velocity v ; the water being then confined between the vanes 
and the curved breast acts by its weight through a distance k % , 




Fig. 101. 

which is approximately equal to h — /i , until finally it is re- 
leased at the level of the tail race and departs with the velocity 
u, which is the same as that of the circumference of the wheel. 
The total energy of the water being Wh, the work of the wheel 
is e Wh, if e be its efficiency. 

The reasoning of the last article may be applied to the 
breast wheel, h x being made equal to zero, and the expression 
there deduced for e may be regarded as an approximate value 
of its theoretic efficiency. It appears, then, that e will be the 
greater the smaller the fall /i ; but owing to leakage between 



Art. 14S.] BREAST WHEELS. 359 

the wheel and the curved breast, which cannot be theoretically 
estimated, and which' is less for high velocities than for low 
ones, it is not desirable to make v and /i small. The efficiency 
of the breast wheel is hence materially less than that of the 
overshot, and usually ranges from 50 to 80 per cent, the lower 
values being for small wheels. 

Another method of determining the theoretic efficiency of 
the breast wheel is to discuss the action of the water in enter- 
ing and leaving the vanes as a case of impulse. Let at the 
point of entrance Av a and An be drawn parallel and equal to 
the velocities v and u, the former being that of the entering 
water and the latter that of the vanes. Let a be the angle 
between v Q and u y which may be called the angle of approach. 
Then the dynamic pressure exerted by the water in entering 
upon and leaving the vanes is, from Art. 133, 

W 
P — — (v cos a — u) , 

and the work performed by it per second is 

W 
k = — (v cos a — tc)u . 

This is a maximum when 

u = $v cos a , 

and the corresponding work of the impulse is 

W 
V=-*o 2 cos 2 *, 

Adding this to the work Wh^ done by the weight of the water, 
the total work of the wheel when running at the advantageous 
velocity is 



k= W\^—--—+/iA; 



360 WA TER WHEELS. [Chap. XIII. 

or if v* be replaced by its' value c? . 2gh Q , where c 1 is the coeffi- 
cient of velocity as determined by the rules of Chapters IV 
and VI, 

k= Wile; cos 2 a. /i + /i 2 ), 

whence the theoretic efficiency is 

£ . h 



X-r* 



.^cosTa^ + J (92) 



If in this expression h 2 be replaced by /i — /i , and if c 1 = 1 
and a = o°, this reduces to the same value as found for the 
overshot wheel. The angle a, however, cannot be zero, for 
then the direction of the entering water would be tangential to 
the wheel, and it could not impinge upon the vanes ; its value, 
however, should be small, say from io° to 25 . The coefficient 
c l is to be rendered large by making the orifice of discharge 
with well-rounded inner corners so as to avoid contraction and 
the losses incident thereto. The above formulas cannot be re- 
lied upon in practice to give close values of k and e, on account 
of losses by foam and leakage along the curved breast, which 
of course cannot be algebraically expressed. 

Prob. 174. A breast wheel is 10.5 feet in diameter, and has 
c 1 =1 0.93, k =4.2 feet, and a = 12 degrees. Compute the 
most advantageous number of revolutions per minute. 



Article 149. Undershot Wheels. 

The common undershot wheel has plane radial vanes, and 
the water passes beneath it in a direction nearly horizontal. 
It may then be regarded as a breast wheel where the action is 
entirely by impulse, so that in the preceding equations // a be- 
comes o, /i becomes h, and a will be o c . The theoretic effi- 
ciencv then is 

e = ic' (92) 



Art. 149.] UNDERSHOT WHEELS. 36 1 

In the best constructions c x is nearly unity, so that it may be 
concluded that the maximum efficiency of the undershot wheel 
is about 0.5. Experiment shows that its actual efficiency varies 
from 0.20 to 0.40, and that the advantageous velocity is about 
0.4^ instead of 0.5ZV The lowest efficiencies are obtained from 
wheels placed in an unlimited flowing current, as upon a scow 
anchored in a stream; and the highest from those where the 
stream beneath the wheel is confined by walls so as to prevent 
the water from spreading laterally. 

The Poncelet wheel, so called from its distinguished in- 
ventor, has curved vanes, which are so arranged that the w T ater 
leaves them tangentially, with its absolute velocity less than 
that of the velocity of the wheel. If in Fig. 101 the fall h^ be 
very small, and the vanes be curved more than represented, it 
will exhibit the main features of the Poncelet wheel. The 
water entering with the absolute velocity v takes the velocity 
u of the vane and the velocity V relative to the vane. Passing 
then under the wheel, its dynamic pressure performs work; and 
on leaving the vane its relative velocity V is probably nearly 
the same as that at entrance. Then if V be drawn tangent to 
the vane at the point of exit, and 71 tangent to the circumfer- 
ence, their resultant will be z\ , the absolute velocity of exit, 
which will be much less than ?/. Consequently the energy car- 
ried away by the departing water is less than in the usual forms 
of breast and undershot wheels, and it is found by experiment 
that the efficiency may be as high as 60 per cent. 

In Fig. 102 is shown a portion of a Poncelet wheel. At A 
the water enters the wheel through a nozzle-like opening with 
the absolute velocity v and at B it leaves with the absolute 
velocity v r In the figure A and B have the same elevation. 
At A the entering stream makes the approach angle a with the 
circumference of the wheel and the same angle with the vane, 
so that the relative velocity V is equal to the velocity of the 



362 



WATER WHEELS. 



[Chap. XIII. 



outer circumference u. If h be the head on A the theoretic 
work of the water is W/i, and the work of the wheel is 



*=*r, 



o> 




Fig. 102. 

and the efficiency, neglecting friction and leakage, is 



2 
C 1 



2^ 

Now, let £, be the coefficient of velocity the entrance orifice, 
then v = c x V2g/i. From the parallelogram's at A and B y 



= 2u sin a 



tan a, 



2 cos a 

and hence the efficiency reduces to 

*= ^ a (i — tan 2 or) (93) 

If c x = I and or = O, the efficiency becomes unity. In the best 
constructions c 1 may be made from 0.95 to 0.98, but a cannot 
be a very small angle, since then no water could enter the 
wheel. l{ a = 30 and c, = 0.95 the efficiency is O.70, which 
is probably a higher value than usually attained in practice. If 
the velocity 71 be greater or less than -A-z^/cos nr, the efficiency 
will be lowered on account of shock and foam at A (Art. 133). 

Prob. 175. Estimate the horse-power that can be obtained 
from a Poncelet wheel under a head of 4 feet, when the orifice 
at A is 2 feet wide and 3 inches deep, taking a = 30 and 
c x = 0.90. 




Art. 150.] VERTICAL IMPULSE WHEELS, 363 

Article 150. Vertical Impulse Wheels. 

A vertical wheel like Fig. 102, but having smaller vanes 
against which the water is delivered from a nozzle, is often 
called an impulse wheel, or a " hurdy-gurdy " wheel. The Pel- 
ton wheel, the Cascade wheel, and other forms, can be pur- 
chased in several sizes, and are convenient on account of their 
portability. Fig. 103 shows an outline 
sketch of such a wheel with the vanes 
somewhat exaggerated in size. The 
simplest vanes are radial planes as at A, 
but these give a low efficiency. Curved 
vanes, as at B, are generally used, as 
these cause the water to turn backward, fl 

opposite to the direction of the motion, B pjlll |[| I II I i I l£ B 
and thus to leave the wheel with a low U 

absolute velocity (Art. 134). In the plan Fig. 103. 

of the wheel it is seen that the vanes may be arranged so as 
also to turn the water sidewise while deflecting it backward. 
The experiments of BROWNE" show that with plane radial 
vanes the highest efficiency was 40.2 per cent, while with 
curved vanes or cups 82.5 per cent was attained. The ve- 
locity of the vanes which gave the highest efficiency was in 
each case almost exactly one-half the velocity of the jet. 

The Pelton wheel is used under high heads, and also being 
of small size it has a high velocity. The effective head is that 
measured at the entrance of the nozzle by a pressure gauge, 
corrected for velocity of approach and the loss in the nozzle by 
formula (39) / . These wheels are wholly of iron, and are pro- 
vided with a casing to prevent the spattering of the water. 
Fig. 104 shows a form with three nozzles, by which three 
streams are applied at different parts of the circumference, in 

* Bowie's Practical Treatise on Hydraulic Mining, p. 193. 



3^4 



WATER WHEELS. 



[Chap. XIII. 



order to obtain a greater power than by a single nozzle, or, by 
using smaller nozzles, in order to obtain a greater speed. For 
an effective head of ioo feet the following quantities are given 




2, 


3j 


4, 


6 


44-19' 


99-52, 


17675, 


39S.08 


363, 


24-> 


1S1, 


121 


7-49> 


16.84, 


29-93, 


67.36 



Fig. 104. 

by the manufacturers for different sizes of Pelton wheels hav 
in^ onlv a single nozzle : 

Diameter in feet, 1, 

Cubic feet per minute, 8.29, 
Revolutions per minute, J26, 
Horse-powers, 1.40, 

which imply an efficiency of 85 per cent. 

The general theory of these vertical impulse wheels is the 
same as that given for moving vanes in Art. 134. Owing to the 
high velocity, more or less shock occurs at entrance, and as the 
angle of exit (3 cannot be made small, the water leaves the 
vanes with more or less absolute velocity. The losses in the 
conduit pipe cannot be fairly charged against the wheel, but 



Art. 150.] VERTICAL IMPULSE WHEELS. 365 

these can be made as small as possible by using a nozzle of 
such diameter as will furnish a stream of maximum horse- 
power. Equation (64) of Art. 85 is a general expression for the 
velocity in the pipe, h being the total head on the nozzle, /the 
length and d the diameter of the pipe, and d x the diameter of 
the nozzle. Let all the resistances except that due to friction 
in the pipe be neglected ; then the velocity of flow from the 
nozzle is expressed by the formula, 



/ 2gh 



V, — — 



V'l 



dr / J , d* 
d* 



in which/" is the friction factor whose mean value is about 0.02. 
Let w be the weight of a cubic foot of water : then the theoretic 
work of the jet per second is 

v* nw( 2ghdd* Y 

and the value of d x which renders this a maximum is, by help 
of the usual method of differentiation, found to be 



4 / d b 



As a special instance, let the pipe be 1200 feet long and one 
foot in diameter: then d x = 0.16 feet = 1.92 inches, and hence 
a 2-inch nozzle is preferable. This result can be revised, if 
thought necessary, by finding the velocity in the pipe and thus 
getting a better value of/* from Table XVI. If the head be 
IOO feet this velocity is found to be 9.4 feet per second, whence 
/= 0.018, and, repeating the computation, d x is 1.97 inches.* 

*See problem by Church in Engineering News, March 1, 1894. 



3 66 



WATER WHEELS. 



[Chap. XIII. 



Prob. 176. The diameter of a hurdy-gurdy wheel is 12.58 
feet between centres of vanes, and the impinging jet has a 
velocity of 58.5 feet per second and a diameter of 0.182 feet. 
The efficiency of the wheel is 44 5 per cent when making 62 
revolutions per minute. What horse-power does it furnish? 



Article 151. Horizontal Impulse Wheels. 

When a wheel is placed with its plane horizontal and is 
driven by a stream of water from a nozzle, as in Fig. 105, it is 
called a horizontal impulse wheel. There are two forms called 
the outward-flow and the inward-flow wheel. In the former the 
water enters the wheel upon the inner and leaves it upon the 




Fig. 105. 

outer circumference, and in the other the reverse is the case. 
The water issuing from the nozzle with the velocity v impinges 
upon the vanes, and in passing through the wheel alters both 
its direction and its absolute velocity, thus transforming its 
energy into useful work. The energy of the entering water is 

IV — , and that of the departing water is IF—, if v. be its abso- 

00- r ° OCT l 

lute velocity. The work imparted to the wheel then is 



W\- 



Art. 151.] HORIZONTAL IMPULSE WHEELS. 367 

and dividing this by the theoretic energy, the efficiency is 



This is the same as the general formula (90) if h' = o, that is, 
if losses in foam and friction are disregarded, and if the wheel 
is set at the level of the tail race. It is now required to state 
the conditions which will render these losses and also the veloc- 
ity v x as small as possible. The reasoning will be general and 
applicable to both outward- and inward-flow wheels. 

At the point A where the water enters the wheel let the 
parallelogram of velocities be drawn, the absolute velocity of 
entrance being resolved into its two components, the velocity 
11 of the wheel at that point, and the velocity ^relative to the 
vane ; let the approach angle between 21 and v be called a, and 
the entrance angle between ?i and V be called <p. At the 
point B where the water leaves the wheel let V x be its velocity 
relative to the vane, and u x the velocity of the wheel at that 
point ; then their resultant is v 19 the absolute velocity of exit. 
Let the exit angle between V x and the reverse direction of u x 
be called /?. The directions of the velocities u and u x are of 
course tangential to the circumferences at the points A and B. 
Let r and r x be the radii of these circumferences ; then the 
velocities of revolution are directly as the radii, or ur l = n x r. 

In order that the water may enter the wheel without shock 
and foam, the relative velocity V should be tangent to the vane 
at A f so that the water may smoothly glide along them. This 
will be the case if the wheel is run at such speed that the 
parallelogram at A can be formed, or whenthe velocities 71 and v 
are proportional to the sines of the angles opposite them in the 
triangle Anv. The velocity z\ will be rendered as small as 
possible by running the wheel at such speed that the velocities 



3 6 8 WATER WHEELS. [Chap. XIIL 

u y and V 1 are equal, since then the parallelogram at B becomes 
a rhombus, and the diagonal v x is a minimum. Hence 

u sin (0 — ol) 

- = : — and u = V. . . . (04) 

v sin l l Ky ^ J 

are the two conditions of maximum efficiency. 

Now, referring to the formula (88) of Art. 136, which ex- 
presses the effect of centrifugal force on the water in revolving 
vanes, it is seen that if u x = V 1 , then also u = V. But u can- 
not equal V unless <fi = 2a, and then it = v/2 cos a, which is 
the advantageous velocity of the circumference at A. There- 
fore the two conditions above reduce to 

v 

d> = 2a, and // = , . . . (q;> 

2 cos a xyDJ 

which show how the wheel should be built and what speed it 
should have to secure the greatest efficiency. When this speed 
obtains the absolute velocity z\ is 

• -1 , r i ■ , * r i sin 2 # 

v = 2u 1 sin ffj = 22i— sin fp = v— -, 

r r cos a 

and the corresponding maximum efficiency is 

;y sin 2 \(5 

e=l r s (96) 

r cos a v J 

by the discussion of which proper values of the angles j3 and a 
can be derived. 

This formula shows that both the approach angle a and the 
exit angle /? should be small in order to give high efficiency, 
but they cannot be zero, as then no water could pass through 
the wheel; values of from 15 to 30 degrees are usual in prac- 
tice. It also shows that /3 is more important than a, and if fi 
be small a may sometimes be made 40 or 45 degrees. 



Art. 152.] DOWNWARD-FLOW WHEELS. 369 

The formula likewise shows that the inward-flow wheel gives 
a higher efficiency than the outward-flow wheel for the same 
value of a and /3, since then r x is less than r. For example, 
let a = 30 , (3 = 30 ; then for an outward-flow wheel where 
r, = 3^ the value of e is 0.196, but for an inward-flow wheel 
where r = ^?\ its value is 0.970. For an outward-flow wheel it 
is hence very necessary that /3 should be a small angle. If the 
entrance angle <p be not made double the approach angle a, the 
efficiency will be lowered by reason of impact at entrance. 

Prob. 177. Compute the maximum efficiency of an out- 
ward-flow impulse wheel when r x = 3 feet, r = 2 feet, a = 45 , 
= 90 , (3 = 30°, and find the number of revolutions per 
minute required to secure such efficiency when the velocity of 
the entering stream is v — 100 feet per second. 



Article 152. Downward-flow Wheels. 

In the impulse wheels thus far considered the water leaves 
the vanes in a horizontal direction. Another form used less 
frequently is that of a horizontal wheel driven by water issuing 
from an inclined nozzle so that it passes downward along the 
vanes without approaching or receding from the axis. Fig. 106 
shows an outline plan of such an impulse wheel and a develop- 
ment of a part of a cylindrical section. Let v be the velocity 
of the entering stream, u that of the wheel at the point where: 
it strikes the vanes, and v x the absolute velocity of the depart- 
ing water. At the entrance A the direction of v makes with 
that of u the approach angle a, and the direction of the relative 
velocity V makes with that of u the entrance angle 0. The 
water then passes over the vane, and, neglecting friction, issues 
at B with the same relative velocity V, whose direction makes 
the exit angle f3 with the plane of motion. 

The condition that impact and foam shall be avoided at A 



3/0 



WA TER WHEELS. 



[Chap. XIII. 



is fulfilled by making V tangent to the vane, and the condition 
that the absolute velocity of exit v, shall be small is fulfilled by 




Fig. 106. 



making the velocities u and V equal at B. Hence, as in the 

last article, the best construction is to make = 2a, and the 
best speed of the wheel is u = v/2 cos a. Also by the same 
reasoning the efficiency under these conditions is 



i — 



sin 2 Ifi, 



cos " a 



which shows that a, and especially /3, should be a small angle 
to give a high numerical value of e. For instance, if both 
these angles are 30 degrees, the efficiency is 0.921, but if a — 
45° and ft = io°, the efficiency is 0.940. 

Although these wheels are but little used, there seems to 
be no hydraulic reason why they should not be employed with 



Art. 153.] SPECIAL FORMS. 37 1 

a success equal to or greater than that attained by vertical im- 
pulse wheels. It will be possible to arrange several nozzles 
around the circumference, and thus to secure a high power with 
a small wheel. The fall of the water through the vertical dis- 
tance between A and B will, also add slightly to the power and 
perhaps also to the resulting efficiency. 

Prob. 178. A wheel like Fig. 106 is to use 2.2 cubic feet of 
water per second, which issues from a nozzle with a velocity of 
IOO feet per second. If the diameter is 3 feet, the efficiency 0.90, 
and the entrance angle = 90 degrees, find the best values of 
the approach and exit angles and the best speed. 

Article 153. Special Forms. 

Numerous varieties of the water wheels above described 
have been used, but the variation lies in mechanical details 
rather than in the introduction of any new hydraulic principles. 
In order that a wheel may be a success it must furnish power 
as cheaply or cheaper than steam or other motors, and to this 
end compactness, durability, and low cost of installation and 
maintenance are essential. 

A variety of the overshot wheel, called the back-pitch 
wheel, has been built in which the water is introduced on the 
back instead of on the front of the wheel. The buckets are 
hence differently arranged from those of the usual form, and 
the wheel revolves also in an opposite direction. 

One of the largest overshot wheels ever constructed is at 
Laxey, on the west coast of England. It is 72^- feet in 
diameter, about 10 feet in width, and is supposed to furnish 
about 150 horse-power, which is used for pumping water out of 
a mine. 

A breast wheel with very long curved vanes extending over 
nearly a fourth of the circumference has been used for small 



372 WA TER WHEELS. [Chap. XIII. 

falls, the water entering directly from the penstock without 
impulse, so that the action is that of weight alone. These are 
made of iron and give a high efficiency. 

Undershot wheels with curved floats for use in the open 
current of a river have been employed, but in order to obtain 
much power they require to be large in size, and hence have 
not been able to compete with other forms. The great amount 
of power wasted in all rivers should, however, incite inventors 
to devise wheels that can economically utilize it. 

The conical wheel, or Danaide, is an ancient form of down- 
ward-flow impulse wheel, in which the water approaches the 
axis as it descends, and thus its relative motion is decreased by 
the centrifugal force. The theory of this is almost precisely 
the same as that of an inward-flow impulse wheel, and there 
seems to be no hydraulic reason why it should not give a high 
efficiency. Another form of Danaide has two or more vertical 
vanes attached to an axis, which are enclosed in a conical case 
to prevent the lateral escape of the water. 

Prob. 179. Consult YVeisbach's Mechanics of Engineering, 
vol. ii., DUBOIS' translation, and discuss the theory of the 
Danaide wheel under a low head, illustrating it by a numericaL 
example. 



Art. 154.J 



THE REACTION WHEEL. 



373 



CHAPTER XIV. 
TURBINES. 

Article 154. The Reaction Wheel. 

The reaction wheel, sometimes called Barker's mill, consists 
of a number of hollow arms connected with a hollow vertical 
shaft, as shown in Fig. 107. The water 
issues from the ends of the arms in a 
direction opposite to that of their 
motion, and by the dynamic pressure 
due to its reaction the energy of the 
water is transformed into useful work. 
Let the head of water CC in the shaft 
be h ; then the pressure-head BB which 
causes the flow from the arms is greater 
than h, on account of centrifugal force. 
Let u 1 be the absolute velocity of the 
exit orifices and V x be the velocity 
of discharge relative to the wheel ; 
then, as shown in Art. 29, and also in 
Art. 13;, 




V x = V2gh + U*. 

The absolute velocity v x of the issuing water now is 



Fig. 107. 



v x = V x — u x = Vzgh -\- 11* — u x . 

It is seen at once that the efficiency can never reach unity 
unless v x = Oj, which requires that V x = u v This, however, can 



374 TURBINES, [Chap. XIV. 

only occur wnen ^ = 00, since the above formula shows that 
V x must be greater than u x for any finite values of h and u x . 
To deduce an expression fo*" the efficiency the work of the 

wheel W\h is to be divided bv the total theoretic work 

o 

W/i, and 

»i' (I\ —u:f 271, 

2gh V * — u* V x + »/ ^' ' 

which shows, as before, that e equals unity when V x = u x = 00 . 
If f^ = 2« lf the value of e is 0.667 ; if V x = $u lt the value of e 

is reduced to 0.50. 

This investigation indicates that the discharge and hence 
the power of a reaction wheel increases with the speed. If a x 
be the area of the exit orifices and w the weight of a cubic unit 
of water, the value of W'vs wa x V x , and when u x = V x = 00 , the 
total work would be utilized, but W would become infinite, as 
also the work of the wheel. Practically, of course, none of 
these conditions can be approached, on account of the losses 
caused by friction. 

To consider the effect of friction in the arms, let c x be the 
coefficient of velocity (Chapter VI), so that 



V x = c x \ 2gk -f-/^ 2 . 
Then the effective work of the wheel is 
W 



k = — fa"i ^2gh + u* — ?/ a 2 ), 



and the efficiency is 



_ c x u x \ 2gh 4- u? — u* 



Art. 154.] THE REACTION WHEEL. 375 

The value of u x , which renders this a maximum, is 

-- gh ,-** 



VI —c 
and this reduces the value of the efficiency to 



e = 1 _ 4/1 _ c x * (98) 

If c v = 1, there is no loss in friction, and u x = 00 and e = I, as 
before deduced. If ^ — 0.94, the advantageous velocity u x is 
very nearly V2gk, and * is 0.66 ; hence the influence of fric- 
tion in diminishing the efficiency is very great. In order to 
make c x large, the end of the arm where the water enters must 
be well rounded to prevent contraction, and the interior sur- 
face must be smooth. If the inner end has sharp square edges, 
as in a standard tube (Art. 61), c x is 0.82, and e becomes 0.43. 

The reaction wheel is not now used as a hydraulic motor on 
account of its low efficiency. Even when run at high speeds 
the efficiency is low on account of the greater friction and re- 
sistance of the air. By experiments on a wheel one meter in 
diameter under a head of 1.3 feet Weisbach found a maxi- 
mum efficiency of 6j per cent when the velocity of revolution 
n x was V^gJi. When u x was 2 \^2gh the efficiency was nothing, 
or all the energy was consumed in frictional resistances. 

The reaction wheel is here introduced at the beginning of 
the discussion of turbines mainly to call attention to the fact 
that the discharge varies with the speed. Although sometimes 
called a turbine, it can scarcely be properly considered as be- 
longing to that class of motors. 

Prob. 180. The sum of the exit orifices of a reaction wheel 
is 4.25 square inches, their radius is 1.75 feet, their velocity 
32.1 feet per second. Compute the head necessary to furnish 
1.6 horse-powers, when c x = 0.9S. 



376 TURBINES. [Chap. XIV. 

Article 155. Classification of Turbines. 

A turbine wheel has been denned as one in which the water 
enters around the entire circumference instead of upon one 
portion, so that all the moving vanes are simultaneously acted 
upon by the dynamic pressure of the water as it changes its 
direction and velocity. Turbines are usually horizontal wheels, 
and like the impulse wheels of the last chapter they may be 
outward-flow, inward-flow, or downward-flow, with respect to the 
manner in which the water passes through them. In the out- 
ward-flow type the water enters the wheel around the entire 
inner circumference and passes out around the entire outer cir- 
cumference (Fig. 109). In the inward-flow type the motion is 
the reverse (Fig. no). In the downward-flow type the water 
enters around the entire upper annular openings, passes down- 
ward between the moving vanes, and leaves through the lower 
annulus (Fig. 115). In all cases the water in leaving the wheel 
should have a low absolute velocity, so that most of its energy 
may be surrendered to the turbine in the form of useful work. 

The supply of water to a turbine is regulated by a gate or 
gates, which can partially or entirely close the orifices where 
the water enters or leaves. The guides and wheel, with the 
gates and the surrounding casings, are made of iron. Numer- 
ous forms with different kinds of gates and different propor- 
tions of guides and vanes are in the market. They are made 
of all sizes from 6 to 60 inches in diameter, and larger sizes are 
built for special cases. The great turbines at Niagara are of 
the outward-flow type, the inner diameter of a wheel being 
63 inches, and each twin turbine furnishing about 5000 horse- 
powers. On account of their cheapness, durability, compact- 
ness, and high efficiency turbines are now more extensively 
used than all other kinds of hydraulic motors. 

The three typical classes of turbines above described are 



Art. 155- ] CLASSIFICATION OF TURBINES. S77 

often called by the names of those who first invented or per- 
fected them ; thus the outward-flow is called the Fourneyron, 
the inward-flow the Francis, and the downward-flow the Jonval, 
turbine. There are also many turbines in the market in which 
the flow is a combination of inward and downward motion, the 
water entering horizontally and inward, and leaving vertically, 
the vanes being warped surfaces. The usual efficiency of tur- 
bines at full gate is from yo to 85 per cent, although 90 per cent 
has in some cases been derived. When the gate is partly 
closed the efficiency in general decreases, and when the gate 
opening is small it becomes very low, as the test in Art. 126 
shows. This is due to the loss of head consequent upon the 
sudden change of cross-section ; and therein lies the disadvan- 
tage of the turbine, for when the water supply is low, it is im- 
portant that the wheel should utilize all the power available. 

Another classification is into impulse and reaction turbines. 
In an impulse turbine the water enters the wheel with a veloc- 
ity due to the head at the point of entrance, just as it does 
from the nozzle which drives an impulse wheel (Art. 151). In a 
reaction turbine, however, the velocity of the entering water 
may be greater or less than that due to the head on the orifices 
of entrance, and, as in the reaction wheel, it is also influenced 
by the speed. This is due to the fact that in a reaction tur- 
bine the static pressure of the water is partially transmitted 
into the moving wheel, provided that the spaces between the 
vanes are fully filled. Any turbine may be made to act either 
as an impulse or a reaction turbine. If it be arranged so that 
the water passes through the vanes without filling them, it is 
an impulse turbine ; if it be placed under water, or if by other 
means the flowing water is compelled to completely fill all the 
passages, it acts as a reaction turbine. As will be seen later the 
theory of the reaction turbine is quite different from that of 
the impulse turbine. 



378 



TURBINES. 



[Chap. XIV. 



Prob. 181. If the efficiency of a turbine is 75 per cent when 
delivering 5000 horse-powers under a head of 136 feet, how 
many gallons of water per minute pass through it? 

Article 156. Reaction Turbines. 

A reaction turbine is driven by the dynamic pressure of 
flowing water which at the same time may be under a certain 
degree of static pressure. If in the reaction wheel of Fig. 107 
the arms be separated from the penstock at A, and be so ar- 
ranged that BA revolves around the axis while AC is station- 
ary, the resulting apparatus may be called a reaction turbine. 
The static pressure of the head CC can still be transmitted 
through the arms, so that, as in the reaction wheel, the dis- 
charge will be influenced by the speed of rotation. The gen- 
eral arrangement of the moving part is, however, like that of 




an impulse wheel, the vanes being set between two annular 
frames, which are attached by arms to a central axis. In Fig. 
108 is a vertical section showing an outward-flow wheel W to 
which the water is brought by guides G from a fixed penstock 



Art. 156.] 



RE A CI' ION TURBINES. 



379 



P. Between the guides and the wheel there is an annular space 
in which slides an annular vertical gate E ; this serves to regu- 
late the quantity of water, and when it is entirely depressed the 
wheel stops. Many other forms of gates are, however, used in 
the different styles of turbines found in the market. 

In Figs. 109 and no are given horizontal and vertical sec- 
tions of both the outward- and the inward-flow types, showing 
the arrangement of guides and vanes. The fixed guide passages 




Fig. 



Fig. 109. 



which lead the water from the penstock are marked G, while 
the moving wheel is marked W. It is seen that the water is 
introduced around the entire circumference of the wheel, and 
hence the quantity supplied, and likewise the power, is far 
greater than in the impulse wheels of the last chapter. 

In order that the static pressure may be transmitted into 
the wheel it is placed under water, as in Fig. 108, or the exit 
orifices are partially closed by gates, or the air is prevented 
from entering them by some other device. 

In Fig, nia Leffel turbine of the inward-flow type is illus- 



[Chap. XIV. 




Fig. 




trated, the arrows showing the 
direction of the water as it 
enters and leaves. The wheel 
itself is not visible, it being with- 
in the enclosing case through 
which the water enters by the 
spaces between the guides. In 
Fig. 112 is shown a view of a 
Hunt turbine, which is also of 
the inward- and downward-flow 
type. In both cases the guides 
are seen with the small shaft for 
moving the gates, these being 
partly raised in Fig. 1 12. The 
flange at the base of the guides 
serves to support the weight of 
the entire apparatus upon the 
floor of the enclosing penstock, 



Art. 156.] 



REACTION TURBINES. 



381 



which is filled with water to the level of the head bay. The 
cylinder below the flange, commonly called a draught-tube, 
carries away the water from the wheel, and the level of the tail 
water should stand a little higher than its lower rim in order to 
prevent the introduction of air, and thus ensure that the wheel 
may act as a reaction turbine. Iron penstocks are frequently 
used instead of wooden ones, and for the pure outward- and 
inward-flow types the wheel is often placed below the level of 
the tail race. 

Turbines are sometimes placed vertically on a horizontal 
shaft. Fig. 113 shows twin Eureka turbines thus arranged in an 
enclosing iron casing. The water enters through a large pipe 




Fig. 113. 

attached to the circular opening, and having filled the cylindri- 
cal casing it passes through the guides, turns the wheels, and 
escapes by the two elbows. Large twin vertical turbines, fur- 
nishing 1200 horse-powers, have been built by the James Leffel 
Company. 

All reaction turbines will act as impulse turbines when from 
any cause the passages between the vanes, or buckets, as they 
are generally called, are not filled with water. In this case the 
theory of their action is exactly like that of the impulse wheels 
described in the last chapter. In Arts. 157-160 reaction tur- 
bines of the simple outward- and inward-flow types will be dis- 



382 



TURBINES. 



[Chap. XIV. 



cussed, the downward-flow type being reserved for special de- 
scription and consideration in Art. 161. 

Prob. 182. Consult Bodmer's Hydraulic Motors, Slagg's 
Water or Hydraulic Motors, and MEISSNER'S Hydraulik, Vol. 
II ; make sketches showing several different arrangements of 
the gates of turbines. 



Article 157. Flow through Reaction Turbines. 

The discharge through an impulse turbine, like that for an 
impulse wheel, depends only on the area of the guide orifices 
and the effective head upon them, or q = av = a V2gk. In a 
reaction turbine, however, the discharge is influenced by the 
speed of revolution, as in the reaction wheel, and also by the 
areas of the entrance and exit orifices. 
To find an expression for this discharge 
let the wheel be supposed to be placed 
below the surface of the tail water, as 
in Fig. 114. Let h be the total head 
between the upper water level and 
that in the tail race, H l the pressure- 
head on the exit orifices and H the 
■ pressure-head at the gate opening as 

Fig. 114. indicated by a piezometer supposed 

to be there inserted. Let «, and u be the velocities of the 
wheel at the exit and entrance circumference, whose radii 
are r, and r (Fig. 109). Let V 1 and V be the relative ve- 
locities of exit and entrance, and v be the absolute velocity 
of the water as it leaves the guides and enters the wheel ; v tt 
may be less or greater than \>2gJi, depending upon the value 
of the pressure-head H. Let a x , a, and a be the areas of the 
orifices normal to the directions of V 1 , V, and r . Now, 
neglecting all losses of friction between the guides, the theorem 




Art. 1 57-] FLOW THROUGH REACTION TURBINES. 383 

of Art. 27, that pressure-head plus velocity-head equals the 
total head, gives the equation 

H+f = h + H x (99) 

Also, neglecting the friction and foam in the buckets, che the- 
orem (89) of Art. 137 gives 

H x +^-~-^ = H+~ . . . . (100) 

1 [ 2g 2g 2g 2g 

Adding these equations, the pressure-heads H x and H disap- 
pear, and there results the formula 

V*-V*-\-v* = 2gk-\-u?-u\ . . . (101) 

Now, since the buckets are fully filled, the same quantity of 
water, q, passes in each second through each of the areas a lf a, 
and a , and hence the three velocities have the respective values, 

K= q -, v= q ~, *. = *-. 

Introducing these values into the formula (101), solving for q, 
and multiplying by a coefficient c to account for losses in leak- 
age and friction, the discharge per second is 



hgk + V? -u 1 ' . 

-^— — (102) 



This is the formula for the flow through a reaction turbine when 
the gate is fully raised. The reasoning applies to an inward- 
flow as well as to an outward-flow wheel. In an outward-flow 
turbine u 1 is greater than u, and consequently the discharge 
increases with the speed ; in an inward-flow turbine u x is less 



384 TURBIXES. [Chap. XIV. 

than u, and consequently the discharge decreases as the speed 
increases. 

The value of the coefficient c will probably vary with the 
head, and also with the size of the areas a lf a, and a . For the 
outward-flow Boyden turbine, the tests of which are given in 
Art. 126, it lies between 0.94 and 0.95, as the following results 
show, where the first four columns contain the number of the 
experiment, the observed head, number of revolutions per 
minute, and discharge in cubic feet per second. The fifth 
column gives the theoretic discharge computed from the above 
formula, taking 1 the coefficient as unity, and the last column is 



No. 


h. 


N. 


1' 


Q- 


c. 


21 


17.16 


63.5 


1 17.01 


123.1 


0.950 


20 


17.27 


70.0 


118.37 


125.2 


0.945 


19 


17.33 


75.0 


119-53 


126.8 


0.943 


18 


17-34 


80.O 


121. 15 


128.4 


0.944 


17 


17.21 


86.O 


122.41 


130.0 


0.942 


16 


17.21 


93-2 


I24.74 


132.5 


0.941 


15 


17.19 


100.0 


127-73 


134.9 


0.947 



derived by dividing the observed discharge q by the theoretic 
discharge Q. The discrepancy of 5 or 6 per cent is smaller 
than might be expected, since the formula does not consider 
frictional resistances. 

A satisfactory formula for the discharge through a turbine 
when the gate is partly depressed is difficult to deduce, because 
the loss of head which then results can only be expressed by 
the help of experimental coefficients similar to those given in 
Art. 75 for the sliding gate in a water pipe, and the values of 
these for turbines are not known. It is. however, certain that 
for each particular gate opening the discharge is given by 



m 



V2gh-\-u x % — u*\ .... (102/ 



Art. 158.] THEORY OF REACTION TURBINES. 385 

in which m depends upon the areas of the orifices and the height 
to which the gate is raised. For instance, in the tests of the 
Boyden turbine of Art. 126, the value of m is 2.815 when the 
proportional gate opening is 0.609, and the computed discharges 
will differ in no case more than one per cent from those 
observed ; when the proportional gate opening is 0.200, the 
value of m is 1.357. And each turbine will have its own values 
of ;//, depending upon the area of its orifices. 

It thus appears that, if the constant in be determined by 
experiment for different gate openings, a reaction turbine may 
be used as a water meter to measure the discharge with a fair 
degree of precision. 

Prob. 183. Consult Francis' Lowell Hydraulic Experi- 
ments, pages 67-75, and compute the coefficient m for ex- 
periments 30 and 31 on the centre vent Boott turbine. 



Article 158. Theory of Reaction Turbines. 

The theory of reaction turbines may be said to include two 
problems: first, given all the dimensions of a turbine and the 
head under which it works, to determine the maximum effi- 
ciency, and the corresponding speed, discharge and power; and 
second, having given the head and the quantity of water to 
design a turbine of high efficiency. This article deals only with 
the first problem, and it should be said at the outset that it 
cannot be fully solved theoretically, even for the best condi- 
tioned wheels, on account of losses in foam, friction, and leak- 
age. The investigation will be limited to the case of full gate, 
since when the gate is partially depressed a loss of energy, due 
to sudden enlargement, generally results (Art. 68). 

The notation will be the same as that used in Chapters XI 
and XIII, and as shown in Figs. 109 and no; the reasoning 



386 TURBINES. [Chap. XIV. 

will apply to both outward- and inward-flow turbines. Let r 
be the radius of the circumference where the water enters the 
wheel and r 1 that of the circumference where it leaves, let u 
and u x be the corresponding velocities of revolution ; then 
ur l = u x r. Let v be the absolute velocity with which the water 
leaves the guides and enters the wheel, and V its velocity of 
entrance ; let a be the approach angle and <p be the entrance 
angle which these velocities make with the direction of u. At 
the exit circumference let V x be the relative velocity with which 
the water leaves the guides, and v x its absolute velocity; let /3 
be the exit angle which V x makes with this circumference. Let 
a Q , a, and a x be the areas of the guide orifices, the entrance, and 
the exit orifices of the wheel, respectively, measured perpen- 
dicular to the directions of v 01 V, and V x . Let d , d, and d x ' : 
the depths of these orifices ; when the gate is fully raised d 
becomes equal to d. 

The areas a , a, a x , neglecting the thickness of the guides 
and vanes, and taking the gate as fully open, have the values 

a = 2 nrd sin a, a = 2rrrd sin 0, .a x = 27tr x d x sin /?; 

and since these areas are fully filled with water, 

q = v . 2nrd sin a = V . 2nrd sin <p = V x . 2nr x d x sin /3. (103) 

These relations, together with the formulas of the last article 
and the geometrical conditions of the parallelograms of veloc- 
ities, include the entire theory of the reaction turbine. 

In order that the efficiency of the turbine may be as high 
as possible the water must enter tangentially to the vanes, and 
the absolute velocity of the issuing water must be as small as 
possible. The first condition will be fulfilled when u and i\ are 
proportional to the sines of the angles (p — a and (p. The second 



Art. 158.] THEORY OF REACTION TURBINES. 387 

will be secured by making u x = V x in the parallelogram at exit, 
as then the diagonal v x becomes very small. Hence 

u sin (0 — a) 

— = ^~ i — . «i = ^i» • • • ( I0 4) 

v Q sin 

are the two conditions of maximum efficiency. 

Now making V t = u x in the third quantity of (103) and 
equating it to the first, there results 

u, rd sin a , u r*d sin a 

and - = , , . . . v io 5) 



v Q r x d x sin p v r 1 *d 1 sin /5* 

Also making V l = u x in (101) and substituting for V 2 its value 
if -f- v* — 2uv {i cos a, from the triangle at A between u and v Qf 
there is found the important relation 

uv cos a = gh, • . (106) 

which gives another condition between u and v . 

Thus three equations between two unknown quantities u and 
v have been deduced for the case of maximumefriciency, namely, 

u sin (0 — a) u r^d sin a gh 

v ~ sin ' v ~ r l i d 1 sin /?' ° ~~ cos a 

If the values of u and v be found from the first and third 
equations, they are 



/gh sin (0 — a) I gh sin , . 

u =\ —. — r> v o = \/ . ,Z — r, (107) 

y cos a sin y cos a sin (0 — a) 

the first of which is the advantageous velocity of the circum- 
ference where the water enters, and the second is the absolute 
velocity with which the water leaves the guides and enters the 
wheel. In order, however, that these expressions may be cor- 



388 TURBIXES. [Chap XIV 

rect, the first and second values of u/v must also be equal, or 

sin (0 — a) r^d sin a 



sin (p r x^x sm ft 



. (108) 



which is the necessary relation between the dimensions and 
angles of the wheel in order that this theory may apply. 

For a turbine so constructed and running at the advantage- 
ous speed, the hydraulic efficiency is 

v* 2u* sin 2 \P t 



e = I ; = I 



2gh gh 

and substituting for ?i l its value in terms of u from (107) and 
having regard to (108), this becomes 

e = I — -j tan a tan \P (109) 

The discharge under the same conditions is q = a v Qf and lastly 
the work of the wheel per second is k = zvqJie. 

The result of this investigation is that the general problem 
of investigating a given turbine cannot be solved theoretically, 
unless it be so built as to approximately satisfy the condition 
in (108). If this be the case it maybe discussed by the formu- 
las deduced. Even then no very satisfactory conclusions can 
be drawn from the numerical values, since the formulas do not 
take into account the loss by friction and that of leakage. To 
determine the efficiency, best speed, and power of a given tur- 
bine, the only way is to actually test it by the method described 
in Art. 126. The above formulas are, however, of great value 
in the discussion of the design of turbines. 

If the coefficient of discharge of a turbine be known (Art. 
157), the advantageous speed and corresponding discharge may 



Art. 159.] DESIGN OF REACTION TURBINES. 389 

be closely computed. For this purpose the condition u l = V l 
= q/a x is to be used. Inserting in this the value of q from 
(102) and solving for u x , there is found 

c 1 . 2srh 
«,' = , , , -, • . • (no) 

which gives the advantageous velocity of the circumference 
where the water leaves the wheel, and then by (102) the dis- 
charge can be obtained. As an example, take the case of 
Holyoke test No. 275, where r x = 2/J- inches, r= 21J inches, 
h = 23.8 feet, a = 2.066, a = 5.526, a 1 — 1.949 square feet, 
a = 25-J , = 90 , (3 = 1 if °. Assuming c = 0.95, as the turbine 
is similar to that investigated in the last article, the formula 
(no) gives u x = 26.61 feet per second, which corresponds to 142 
revolutions per minute, which agrees well with the actual num- 
ber, 138. The efficiency found by the test at that speed was 
0.79, which is a very much less value than the above theoretic 
formulas will give. 

Prob. 184. For the case of the last problem r = 4.67, r 1 = 
3.95, ^=1.01, ^ — 1.23, A =13.4 feet, ^ = 9 °.5, 0=ii 9 °, 
/3 = n°. Compute the areas a of a, a lf and the advantageous 
speed. 

Article 159. Design of Reaction Turbines. 

The design of an outward- or inward-flow turbine for a given 
head and discharge includes the determination of the dimen- 
sions r, r a , d, d x , and the angles a, /?, and 0. These may be 
selected in very many different ways, and the formulas of the 
last article furnish a guide how to do this so as to secure a high 
degree of efficiency. 

First, it is seen from (109) that the approach angle a and 



390 TURBIXES. [Chap. XIV. 

the exit angle should be small, but that, as in other wheels, 
ft has a greater influence than a. However, ft must usually be 
greater for an inward-flow than for an outward-flow wheel in 
order to make the orifices of exit of sufficient size. For the 
entrance angle cp a good value is 90 degrees, and in this case 
the velocity u is always that due to one-half the head, as seen 
from (107). The radii r and r 1 should not differ too much, as 
then the frictional resistance of the flowing water and the 
moving wheel would be large. It is also seen that the effi- 
ciency is increased by making the exit depth d 1 greater than 
the entrance depth d, but usually these cannot greatly differ, 
and are often taken equal. 

Secondly, it is seen that the dimensions and angles should 
be such as to satisfy the formula (108), since if this be not the 
case losses due to impact at entrance will occur which will ren- 
der the other formulas of little value. 

As a numerical illustration let it be required to design an 
outward-flow reaction turbine which shall use 120 cubic feet per 
second under a head of 18 feet and make 100 revolutions per 
minute. Let the entrance angle (J) be taken at 90 degrees, then 
from (107) the best velocity of the inner circumference is 



u = X 32.16 X 18 = 24.06 feet per second, 

and hence the inner radius of the wheel is 

60 X 24.06 
r = ■ = 2.298 feet. 

2,T X IOO J 

Now let the outer radius of the wheel be three feet, and also 
let the depths ^/and d x be equal ; then from (108) 

sin ft /2.2q8\ 2 „■ 

= — =0.5866. 

tan a V3.000/ 

If the approach angle a be taken as 30 degrees, the value of the 



Art. 159.] DESIGN OF REACTION TURBINES. 39 1 

exit angle /3 to satisfy this equation is 19 48', and from (109) 
the hydraulic efficiency is 0.899. If, however, a be 24 degrees, 
the value of (3 is 15 08' and the hydraulic efficiency is 0.941 ; 
these values of a and ft will hence be selected. 

The depth d is to be chosen so that the given quantity of 
water may pass out of the guide orifices with the proper veloc- 
ity. This velocity is, from (107), 

v = 24.06/cos 24 = 26.34 feet per second ; 
and hence the area of the guide orifices should be 

a = 120/26.34 = 4.556 square feet, 
from which the depth of the orifices and wheel is 

d— 4.5 56/2 7zr sin 24 = 0.776 feet. 

As a check on the computations the velocities Fand V lt with 
the corresponding areas a and a ot may be found, and d be again 
determined in two ways. Thus, 

V = v sin 24 =10.71, V x = u x = urjr = 31.42 ft. per sec. ; 
a — 120/10.71 = 11.204, a i — 120/31.42 = 3.820 square ft. ; 
d = 1 1.204/ 27tr = 0.776, d x = 3.820/2^^ sin f3 = 0.776 feet ; 

and this completes the preliminary design, which should now 
be revised so that the several areas may not include the thick- 
ness of the guides and vanes (Art. 160). 

Although the hydraulic efficiency of this reaction/turbine is 
94 per cent, the practical efficiency will probably not exceed 80 
per cent. About 2 per cent of the total work will be lost in 
axle friction. The losses due to the friction of the water in 
passing through the guides and vanes, together with that of the 
wheel revolving in water, and perhaps also a loss in leakage, 
will probably amount to more than one-tenth of the total 



39 2 TURBINES. [Chap. XIV. 

work. All of these losses influence the economic velocity so 
that a test would be likely to show that the highest efficiency 
would obtain for a speed less than ioo revolutions per minute. 

Prob. 185. Design an inward-flow reaction turbine which 
shall use 120 cubic feet per second under a head of 18 feet 
while making 100 revolutions per minute, taking <p = 68°, 
a- iO°, and p = 21°. 

Article 160. Guides and Vanes. 

The discussions in the last two articles have neglected the 
thickness of the guides and vanes. As these, however, occupy 
a considerable space a more correct investigation will here be 
made to take them into account. Let t be the thickness of a 
guide and n their number, t x the thickness of a vane and n x their 
number. Then the areas a Q , a, and a l perpendicular to the 
directions of v , V, and l\, are strictly 

a Q = (27tr sin a — ni\d, a = (2,71 r sin — n x t^)d, 
a x = {2?rr 1 sin /3 — ^O^i » 

and the expressions in (103) are 

q = a v =aV=a 1 V l , . . . . (103)' 

while those in (105) become 



^0 ~ a, ' v % ~ a x r; 

also, the necessary condition in (10S) is 

sin (d> — a) a Q r 
sin <p ~ a x r x 

and the greatest hydraulic efficiency is given by 



(105)' 



(108)' 



r? sin(0— a) sin 2 ±0 . v 

I — 2~ )—- , . . . (109)' 

r sin Q cos a 



Art. 160.] GUIDES AND VANES. 393 

in which, of course, sin (0 — «)/sin <p may be replaced by its 
equivalent a Q r/a x r v The advantageous speed is, as before, 
given by (107). 

To discuss a special case, let the example of the last article 
be again taken. An outward-flow turbine is to be designed to 
use 120 cubic feet of water under a head of 18 feet while 
making 100 revolutions per minute, the gate being fully 
opened. The preliminary design has furnished the values 
r = 2.298 feet, r — 3.000 feet, d ' = d x = 0.776 feet, <p — 90 , 
a = 24 , /3 = 1 5° 08'. It is now required to revise these so 
that 24 guides and 36 vanes may be introduced. Each of 
these will be made one-half an inch thick, but on the inner cir- 
cumference of the wheel the vanes will be thinned or rounded 
so as to prevent shock and foam that might be caused by the 
entering water striking against their square ends (see Fig. 122). 
If the radii and angles remain unchanged the effect of the 
vanes will be to increase the depth of the wheel, which is now 
0.702 feet wide and 0.776 feet deep. As these are good pro- 
portions, it will perhaps be best to keep the depth and the radii 
unchanged, and to see how the angles and the efficiency will 
be affected. 

Since the vanes are to be thinned at the inner circumference 
the area a is unaltered and its value is simply 2nrd sin 0. 
Hence remains 90 degrees, and V is unchanged. This re- 
quires that the area a should remain the same as before. The 
area a 1 is also the same, as its value is q/u^ Accordingly 

4.556 = (2rtr sin a — 2^t)d, 3.820 = {2nr sin j3 — 36/^, 

in which a and (3 are alone unknown. Inserting the numerical 
values and solving, a = 28 26 r and /3 = 19 5-5', both being in- 
creased by about 4J degrees. From (109/ the efficiency is now 
found to be 0.898, a decrease of 0.043, due to the introduction 
of the guides and vanes. 



394 



TURBINES. 



[Chap. XIV. 



The efficiency may be slightly raised by making the outer 
depth d 1 greater than the inner depth d. For instance, let 
d 1 = 0.816, while d remains 0.776 ; then ft is found to be 19 
06', and e = 0.906. But another way is to thin down the vanes 
at the exit circumference and thus maintain the full area a 1 
with a small angle ft. If this be done in the present case d 1 
may be kept at 0.776 feet, ft be reduced to about 16 degrees, 
and the efficiency will then be about 0.92 or 0.93. 

No particular curve for the guides and vanes is required, 
but it must be such as to be tangent to the circumferences at 
the designated angles. The area between two vanes on any 
cross-section normal to the direction of the velocity should also 
not be greater than the area at entrance ; in order to secure 
this vanes are frequently made much thicker at the middle than 
at the ends (see Fig. 122). 

Prob. 186. Find the best speed and the probable dis- 
charge and power of the turbine designed above when under a 
head of 50 feet. 



Article 161. Downward-flow Turbines. 

Downward- or parallel-flow turbines are those in which the 
water passes through the wheel without changing its distance 



w. ; y y 



Fig. 115. 

from the axis of revolution. In Fig. 115 is a semi-vertical sec- 
tion of the guide and wheel passages, and also a development 
of a portion of a cylindrical section showing the inner arrange- 



Art. 161.] 



DO WNWARD-FLO W TURBINES. 



395 



merit. The formula for the discharge can be adapted to this 
by making u x = u. In this turbine there is no action of centrif- 
ugal force, so that the relative exit velocity V l is equal to V, 

The great advantage of this form of turbine is that it can 
be set some distance above the tail race and still obtain the 
power due to the total fall. This distance cannot exceed 34 
feet, the height of the water barom- 
eter, and usually it does not ex- 
ceed 25 feet. Fig. 116 shows in a 
diagramatic way a cross-section of 
the penstock P, the guide pas- 
sages G, the wheel W, and the air- 
tight draught tube T, from which 
the water escapes by a gate E to 
the tail race. The pressure-head 
H x on the exit orifices is here 
negative, so that the air pressure 
equivalent to this head is added to 
the water pressure in the pen- 
stock, and hence the discharge 
through the guides occurs as if 
the wheel were set at the level of 
the tail race. Strictly speaking a 
vacuum, more or less complete, is FlG - II6 - 

formed just below the wheel into which the water drops with 
a low absolute velocity, having surrendered to the wheel nearly 
all its energy. Short draught tubes are also often used with 
inward-flow turbines. 

Let h be the total head between the water levels, in the 
head and tail races, h Q the depth of the entrance orifices of the 
wheel below the upper level, h x the vertical height of the wheel, 
and h % the height of the exit orifices above the tail race ; so 




396 TURBINES. [Chap. XIV. 

that h = /i + ^1 + K' Let J7 and H 1 be the heads which 
measure the absolute pressures at the entrance and exit orifices 
of the wheel and h a the height of the water barometer. Let v 
be the absolute velocity with which the water leaves the guides 
and enters the vanes, and V and J\ the relative velocities at 
entrance and exit. Then from Article 27 

v: = 2 g {h a +h a -H), 

Adding these two equations there results 

v.* -v>+ v; = 2g(k, + *, + k - H t \ 

But h a — H l is equal to h %1 and hence 

Vt '-V'+K'=2^ (IOI)' 

This formula is the same as (101), if u be made equal to u l3 and 
hence all the formulas of the last three articles apply to the 
downward-flow reaction turbine by placing u = « lf and r = r lB 

Let r be the mean radius and u the mean velocity of the 
entrance and exit orifices of the wheel, let d be the width of 
the entrance orifices and d 1 that of the exit orifices. Let a be 
the approach angle which the direction of the entering water 
makes with that of the velocity #, or the angle which the guides 
make with the upper plane of the wheel (Fig. 116); let £> be 
the entrance angle which the vanes make with that plane, and/5 
the acute exit angle which they make with the lower plane. 
Then the values of the advantageous velocity u and the enter- 
ine velocity v n are 



y/- 



o;h sin (<p — a) / gh sin o . x/ 

, S r o = A / —- ^ i' U°7) 

cos a sm V cos a sm {p — a) 



Art. 162.] IMPULSE TURBINES. 397 

and the necessary relation between the angles and dimensions 
of the wheel is 



sin (0 — a) d sin a a c 
sin <fi ~ d x sin /3 ~~ a^ 

while the hydraulic efficiency is 



(10! 



a sin 2 -J/? d 

e — 1 — 2 = 1 : tan a tan \B. . (iooV 

a x cos a d x ' x ^ 

To these equations is to be added the condition that the 
pressure-head H 1 cannot be less than that of a vacuum, and on 
account of air leakage it must be practically greater ; thus 

that is, the height of the wheel orifices above the tail race must 
be less than the height of the water barometer. 

As an example of design, let cp — 90 and a = 30 . Then 
u := ^ gh, or the velocity due to one-half the head ; and 
v o = v^gh, or a velocity due to two-thirds of the head. From 
(io8y, taking d x — -|4 the value of j3 is 21° 24/, and from (109)' 
the efficiency is 0.93. This value will be lowered by the intro- 
duction of guides and vanes, as well as by friction and the. energy 
carried away by the water as it escapes through the gate into 
the tail race, so that perhaps not more than 0.80 will be ob 
tained in practice. 

Prob. 187. A downward-flow turbine has d= d x , a = 16 , 
/3 = 1 5°, h= 50 feet; compute the angle <fi, the best speed, 
and the hydraulic efficiency. 

Article 162. Impulse Turbines. 

Whenever a turbine is so arranged that the channels be- 
tween the vanes are not fully filled with water, it ceases to act 
as a reaction turbine and becomes an impulse turbine. A tur- 



39§ TURBINES. [Chap XIV, 

bine set above the level of the tail race becomes an impulse 
turbine when the gate is partially lowered, unless the gates are 
arranged over the exit orifices. 

The velocity with which the water leaves the guides in an 
impulse turbine is simply V2gh where 7i tj is the head on the 
guide orifices. The rules and formulas in Art. 151 apply in 
all respects, and for a well-designed wheel the entrance angle 
is double the approach angle a, the best speed and corre- 
sponding hydraulic efficiency are 



~ V 2 cos 2 a ' \rcosa)' 



while the discharge is q = a V2gh Q and the work per second 
is k = wqh^e. 

As an example, suppose that the reaction turbine designed in 
Art. 159 were to act as an impulse turbine, the angles a and ft 
remaining at 24 and 15 08', the radii r and r 1 being 2.298 and 
3.000 feet. It would then be necessary that <p should be 48 
instead of 90 in order to secure the best results. Under a 
head of 18 feet the velocity of flow from the guides would be 
24.03 feet per second instead of 26.34. The velocity of the 
inner circumference would be 18.20 feet per second instead of 
34.06, so that the number of revolutions per minute would be 
about 75 instead of 100. The efficiency would be 0.94, or 
almost exactly the same as before. If, however, the angle 
were to remain 90 the efficiency would be materially lowered, 
since then the water could not enter tangentially to the vanes 
and a loss in impact would necessarily result. 

Impulse turbines move slower than reaction turbines under 
the same head, but the relative entrance velocity /""is greater, 
and hence more energy is liable to be spent in shock and foam. 



Art. 163.] SPECIAL DEVICES. 399 

In impulse turbines the entrance angle should be double the 
approach angle a, but in reaction turbines it is often greater 
than 3^, and its value depends upon the exit angle/?: hence 
the vanes in impulse turbines are of sharper curvature for the 
same values of a and /3. In impulse turbines the efficiency is 
not lowered by a partial closing of the gates, whereas the 
sudden enlargement of section causes a material loss in reaction 
turbines. The advantageous speed of an impulse turbine re- 
mains the same for all positions of the gate, but with a reaction 
turbine it is very much slower at part gate than at full gate. 
For many kinds of machinery it is important to maintain a con- 
stant speed for different amounts of power, and with a reaction 
turbine this can only be done by a great loss in efficiency. 
When the water supply is low the impulse turbine hence has a 
marked advantage in efficiency. A further merit of the im- 
pulse turbine is that it may be arranged, so that water enters 
only through a part of the guides, while this is impossible in re- 
action turbines. On the other hand, reaction turbines can be 
set below the level of the tail race or above it, using a draught 
tube in the latter case, and still secure the power due to the 
total fall, whereas an impulse turbine must always be set above 
the tail-race level and loses all the fall between that level and 
the guide orifices. 

Prob. 188. Compare the advantageous speeds of impulse 
and reaction turbines when the velocity of the water issuing 
from the guide orifices is the same. 

Article 163. Special Devices. 

Many devices to increase the efficiency of reaction turbines, 
particularly at part gate, have been proposed. In the Four- 
neyron turbine a common plan is to divide the wheel into three 
parts by horizontal partitions between the vanes so that these 
are completely filled with water when the gate is either one- 



400 



TURBINES. 



[Chap. XIV. 



third or two-thirds closed (see Fig. 121). The surface exposed 
to friction is thus, however, materially increased at full gate. 

The Boyden diffuser is another device used with outward- 
flow reaction turbines. This consists of a fixed wooden annu- 




FlG. 117. 

lar frame D placed around the wheel IV, through which the 
water must pass after exit from the wheel. Its width is about 
four or five times that of the wheel, and at the outer end its 
depth becomes about double that of the wheel. The effect 
of this is like a draught tube, and although the absolute velocity 
of the water when issuing from the wheel is greater than be- 
fore, the absolute velocity of the water coming out of the 
diffuser is less, and hence a greater amount of energy is im- 
parted to the turbine. It has been shown above that the 
efficiency of a reaction turbine is increased by making the exit 
depth d x greater than the entrance depth d, and it is seen that 
the fixed diffuser produces the same result. By the use of this 
diffuser BOYDEN increased the efficiency of the Fourneyron re- 
action turbine several per cent. 

The pneumatic turbine of GlRARD was devised to overcome 
the loss in reaction turbines due to a partial closing of the gate. 
The turbine was inclosed in a kind of bell into which air could 
be pumped, thus lowering the tail water level around the wheel. 
At part gate this pump is put into action, and as a consequence 
the air is admitted into the wheel, and the water flowing 
through it does not fill the spaces between the vanes. Hence 



Art. 163.] SPECIAL DEVICES. 401 

the action becomes like that of an impulse turbine, and the full 
efficiency is maintained. A wheel thus arranged should prop- 
erly have the entrance angle <p double the approach angle a in 
order that the advantageous speed may be always the same. 

Turbines without guides have been used. Here the ap- 
proach angle a is probably about 90 degrees, as the water 
would probably approach the wheel by the shortest path. The 
entrance angle <p would then be made greater than 90 degrees, 
and the reliance for high efficiency must be upon a small value 
of the exit angle /?. But as this can scarcely be made smaller 
than 15 degrees the hydraulic efficiency will rarely exceed 80 
per cent, which by friction and foam will in practice be reduced 
to about 65 per cent. 

The screw turbine consists of one or two turns of a heli- 
coidal surface around a vertical shaft, the screw being enclosed 
in a cylindrical case. At the point of entrance the downward 
pressure of the water can be resolved into two components, a 
relative velocity V parallel to the surface and a horizontal 
velocity u which corresponds to the velocity of the wheel. At 
the point of exit it can be resolved in like manner into V 1 and 
U x . But, as in other cases, the condition for high efficiency is 
u x = V l9 and since the water moves parallel to the axis, u x = u. 
Applying the general formulas (99)-(ioi), it is seen that this 
can only occur when the head h is zero or when the velocity u 
is infinite. The screw turbine is hence like a reaction wheel, 
and high efficiency can never practically be obtained. 

Prob. 189. Consult RtJHLMANN's Maschinenlehre, Vol. I, pp. 
360-425, and describe a scheme for "ventilating" a turbine in 
order to increase its efficiency. 



402 TURBINES. [Chap. XIV. 

Article 164. The Niagara Turbines. 

A number of turbines have been installed at Niagara for 
the utilization of a portion of the power of the great falls. 
Those to be here briefly described are the three large wheels 
designed bv Faesch and Picard, of Geneva, Switzerland, and 
erected in 1894 and 1895 by the Cataract Construction Com- 
pany. The entire plant is to include ten twin outward-flow 
reaction turbines, each of 5000 horse-power. 

The power plant is located about 1 \ miles above the village 
of Niagara Falls, where a canal leads the water from the river 
to the wheel pit. The water is carried down the pit through 
steel penstocks to the turbines, which are placed 136 feet below 
the water level in the canal. After passing through the wheels 
the waste water is conveyed to the river below the American 
fall by a tunnel 7000 feet long. 

Fig. 118 shows part of a longitudinal section of the wheel 
pit and a side view of two of the penstocks, with the enclosing 
cases and shafts of the turbines. Fig. 119 shows a cross-sec- 
tion of the wheel pit with an end view of a penstock, wheel 
case, and shaft. The width of the wheel pit is 20 feet at the 
top and 16 feet at the bottom, and the cylindrical penstock is 
7J- feet in diameter. The shaft of the turbine is a steel tube 
38 inches in diameter, built in three sections, and connected by 
short solid steel shafts 11 inches in diameter which revolve in 
bearings. At the top of each shaft is a dynamo for generating 
the electric power. 

In Fig. 120 is shown a vertical section of the lower part of 
the penstock, shaft, and twin wheels. The water fills the casing 
around the shaft, passes both upward and downward to the 
guide passages, marked G, through which it enters the two 
wheels, causes them to revolve, and then drops down to the tail 



Art. 164.] 



THE NIAGARA TURBINES. 

ram xtm 



403 







Center line ofj[urbines ^§C. 



, _ J L A^ _ _ _ A^f S 

rr 11 nil 1 1 1 1 fill 11 it 




$^?^^^^$&rj£ 



Bradley $ Puatet., Engr'.a, N.F. 



Fig. ii 



404 



TURBINES. 



[Chap. XIV. 







(Tenter Lrr _ / _ _'_: ~~~ EEj 




Art. 164." 



THE NIAGARA TURBINES. 



405 



race at the entrance to the tunnel, which carries it away to the 
river. The gate for regulating the discharge is seen upon the 
outside of the wheels. 




Fig. 120. 

Fig. 121 gives a larger vertical section of the lower wheel 
with the guides, shaft, and connecting members. The guide 
passages, marked G, and the wheel passages, marked W, are 
triple, so that the latter may be rilled not only at full gate, 
but also when it is one-third or two-thirds opened, thus avoid- 
ing the loss of energy due to sudden enlargement of the 
flowing stream. The two horizontal partitions in the wheel are 
also advantageous in strengthening it. The inner radius of the 
wheel is 3 ii inches and the outer radius is 37J inches, while the 
depth is about 12 inches. In this figure the gates are repre- 
sented as closed. 



406 



TURBINES. 



[Chap. XIV. 



In Fig. 122 is shown a half-plan of one of the wheels, on a 
part of which are seen the guides and vanes, there being 36 of 
the former and 32 of the latter. The value of the approach 
angle a is 19 06', the mean value of the entrance angle is 
iio° 40', and the exit angle /3 is 13 i/i'. Although the water 




Fig. i2i. 

on leaving the wheel is discharged into the air, the very small 
annular space between the guides and vanes, together with the 
decreasing area between the vanes from the entrance to the 
exit orifices, will ensure that the wheels will act like reaction 
turbines for the three positions of the gates corresponding to 
the three horizontal stages. 

The estimated discharge of one of these twin turbines is 
about 430 cubic feet per second, and the theoretic power due 
to this discharge is 6645 horse-powers. Hence if 5000 horse- 
powers be utilized the efficiency is 75.2 per cent. Under this 
discharge the mean velocity in the penstock will be nearly 10 
feet per second, but the loss of head due to friction in the pen- 
stock will be but a small fraction of a foot. The pressure- 



A; t. 164.J 



THE NIAGARA TURBINES. 



407 



head in the wheel case will then be practically that due to the 
actual static head, or closely 141^- feet upon the lower and 130 
feet upon the upper wheel. Although the penstock is smaller 
in section than generally thought necessary for such a large 
discharge, the loss of head that occurs in it is insignificant ; 




Fig. 



and it will be seen in Fig. 118 to be connected with the head 
canal and with the wheel case by easy curves, and that its 
section is enlarged in making these approaches. 

From formula (107) the advantageous velocity of the inner 
circumference of the upper wheel, taking h = 130-J feet, is 



408 TURBINES. [Chap. XIV. 

found to be 64.47 ^ ee ^ P er second, and that for the lower wheel, 
taking h = 141^ feet, is found to be 71.73 feet per second. 
Perhaps the mean of these, or 68. 10 feet per second, will closely 
correspond with the advantageous velocity for the two com- 
bined. The number of revolutions per minute for the condition 
of maximum efficiency will then be closely 250. The absolute 
velocity of the water when entering the wheel will be about 64 
feet per second, so that the pressure-head in the guide passages 
of the upper wheel will be nearly 66 feet. The mean absolute 
velocity of the water when leaving the wheels is about 19 feet 
per second, so that the loss due to this is only about 4 per cent 
of the total head. 

The weight of the dynamo, shaft, and turbine is carried, 
when the wheels are in motion, by the upward pressure of the 
water in the wheel case on a piston placed above the upper 
wheel. The upper disk containing the guides is, for this pur- 
pose, perforated, so that the water pressure can be transmitted 
through it. The lower disk, on the other hand, is solid, and 
the weight of the water upon it is carried by inclined rods up- 
ward to the wheel case, which together with the penstock is 
supported upon several girders. At the upper end of the shaft 
is a thrust bearing to receive the excess of vertical pressure, 
which may be either upward or downward under different con- 
ditions of power and speed. 

A governor is provided for the regulation of the speed, and 
this is located on the surface near the dynamo. It is of the 
centrifugal-ball type, and so connected with the main shaft and 
the turbine gates that the latter are partially closed whenever 
from any cause the speed increases. These gates are so set 
that the orifices of the upper and lower wheels are not simul- 
taneously closed, one gate being in advance of the other by 
about the width of one division stage. The revolving field 



Art. 164.] THE NIAGARA TURBINES. 409 

magnets of the dynamo also serve as a fly-wheel for equalizing 
the speed. With this method of regulation it is expected that 
the speed cannot increase more than three or four per cent 
when 25 per cent of the work is suddenly removed. 

These turbines were designed by the Swiss engineers above 
named, after an international competition in which three Swiss, 
one Austrian, four French, three English, and two American 
firms participated. Three turbines have been built by I. P. 
MORRIS & COMPANY, of Philadelphia, and the installation of 
these was completed early in 1895.* 

Prob. 190. Compute the hydraulic efficiency of these wheels, 
neglecting losses due to friction and foam. Compute also the 
velocity with which the water enters and leaves the wheel when 
running at the advantageous speed. 

*See Engineering News, Jan. ^3, 1892, and March 30, 1893 ; also an article 
by Herschel in Cassiers' Magazine for March, 1893. For many of the above 
facts the author is indebted to the kindness of the officers of the Cataract Con- 
struction Company, through Dr. Coleman Sellers, President and Chief 
Engineer of the Niagara Falls Power Company, 



4IO APPENDIX. 



APPENDIX. 



Answers to Problems. 

Below will be found the answers to most of the problems 
whose solution is not stated in the text, the number of the 
problem being enclosed in parenthesis. A few answers have 
been purposely omitted in order that the student may be 
thrown entirely upon his own resources. However satisfac- 
tory it may be to know in advance the result of the solution 
of an exercise, let the student bear in mind that after com- 
mencement day answers to problems will not be given him. 

Chapter I. (i), 996, or, in round numbers, 1000 kilos per 
square centimeter. (3), 393 pounds, 47.1 gallons, 178 kilos. 
(5), 73.8 pounds, 5.02 atmospheres. (9), 14.73 pounds. (10), 
7.85 gallons, 65.6 pounds. 

Chapter II. (12), 100 feet, 100 meters. (16), 5280 pounds. 
(17), 2880 feet. (18), 8.04 feet, 2000 pounds. (19), y = h.L 
(22), S.y feet and 8.4 feet. (23), 8.7 pounds. 

Chapter IIL (27), v = 8.75 and 2.19 feet per second. 
(29), 0.0534 cubic feet per second. (33), 15.3 and 19.6 cubic 
feet per second. (34), 90.1 feet per second. (35), 94.6 feet 
per second. (37), -f \ 2 times that for the hemisphere. (39), 
318. (42), 1.8 1 horse-powers. 



ANSWERS TO PROBLEMS: 411 

Chapter IV. (47), 24 feet. (48), 0.617. (49), 0.981. (50), 
11.97. (52), 0.601. (53), 0.0169 cubic feet per second. (54), 
O.605. (55), 5.85 cubic feet per second. (56), 17.3 feet. (58), 
101. (59), 2.04 cubic feet. (60), 6.05 cubic feet per second. 
(61), 0.88. (62), 0.18 of one per cent. (64), 4 minutes, 13 sec- 
onds. (65), 9.4 and 12.3 square feet. 

Chapter V. (66), 2 feet per second. (68), 0.00173 feet. 
(69), 0.837 f eet P er second, and 0.0109 feet. (70), 1.45 1, 1.458, 
1.465 cubic feet per second. (73), 4.035 cubic feet per second. 
(74), 7.10 and 6.97 cubic feet per second. (75), 21.1 cubic feet 
per second. (y6), 1.82 feet, (yy), 7.58 feet. ' (78), if h — o, 
then strictly H —- o. (79), 0.74 per cent. 

Chapter VI. (80), 5.69 and 5.46 horse-powers. (81), 0.985. 
(82), 0.962. (84), — 2.89 pounds, 6.67 feet. (85), 0.802. (88), 
0.13 and 7.60 feet. (89), 0.28 feet. (91), 0.995. 

Chapter VII. (92), 0.135. (93), 0.29 feet. (94), 7.64 and 
8.44 feet. (95), 0.26 feet. (96), 3.39 feet per second. (98), 
6.14 gallons per minute. (100), 3.06 and 4.80 inches. (101), 
26 gallons per minute. (103), 0.935 and 0.722 feet. (104), 32. 
(105), 6. 13 cubic feet per second. (108), 0.82 inches, (no), 
2.75 feet per second, 68 feet. (112), 17.8 horse-powers. (113), 
O.036. (114), 15.8 cubic feet per second, 0.913 and 0.705 feet. 
(115), 56440. 

Chapter VIII. (116), 1.1 foot. (1 17), 2.54 feet per second. 
(119), 226.5 cubic feet per second. (120), 4.4 feet. (121), 1. 214 
feet and 7.13 feet per second. (122), 1 : 1.21. (123), 0.64 feet 
deep. (124), 5.20 and 3.69 feet per second. (125), 61 300 000. 
(126), 64750000. (127), 57630000. (128), 4.8 feet. (131), 
0.81 horse-powers. 



412 APPENDIX. 

Chapter IX. (133), 547 cubic feet per second. (134), 364 
pounds. (135), 4.85 feet per second. (137), 0.81 feet per sec- 
ond. (139), 1.59 feet per second. (142), 770 cubic feet per 
second. (145), 4-5 feet at one mile. 

Chapter X. (146), 1.28 horse-powers. (147), 0.00266. (148), 
2.57 feet. (149), 0.32. (150), 0.488 horse-powers, and 35.4 per 
cent efficiency. 

Chapter XI. (154), 3-97- (i55)> 32.1 pounds. (158), 93.2 
pounds. (160), 34.5 feet per second. (161), 507. (162), effi- 
ciency is 85.1 per cent. (163), 0.83. 

Chapter XII. (164), 743. (165), 259 pounds. (166), 1530. 
(167), velocity less than four knots per hour and efficiency less 
than 0.10. (168), 56 feet in diameter. (171), 8.34, 7.71. 6.86, 
and 5. 1 1 feet per second. 

Chapter XIII. (172), 1.04 horse-powers. (173), about 49 
horse-powers. (174), 27.2. (175), about 3.5 horse-powers. 
(176), 4.09 horse-powers. (177), 275 revolutions per minute. 
(178), /3 = 26 degrees. 

Chapter XIV. (180), 16 feet. (181), 193620 gallons per 
minute. (183), 4. 117 and 4.120. (185), see the article Hydro- 
dynamics in the Encyclopaedia Britannica. (186), 167 revolu- 
tions per minute. (187), <fi — no° 40'. 



DESCRIPTION OF TABLES. 413 



Description of Tables. 

Table XXVI, on the next two pages, gives four-place 
logarithms of numbers which will be found very useful and 
sufficiently accurate for most hydraulic computations. 

Table XXVII gives squares of numbers from 1.00 to 9.99, 
the arrangement being the same as that of the logarithmic 
table. By moving the decimal point four-place squares of other 
numbers are also readily taken out. For example, the square 
of 0.874 is 0.7639, that of 87.4 is 7639, and that of 874 is 763 900, 
correct to four significant figures. 

Table XXVIII gives areas of circles for diameters ranging 
from 1. OO to 9.99, arranged in the same manner, and by prop- 
erly moving the decimal point four-place areas for all circles 
can be found. For instance, if the diameter is 4.175 inches, 
the area is 13.69 square inches ; if the diameter is 0.535 f eet > the 
area is 0.2248 square feet. 

Table XXIX gives the loss of head in friction in clean iron 
pipes, either smooth or coated, and laid with good joints. The 
friction-heads are given for a pipe 100 feet long and for differ- 
ent velocities ranging from 1 to 15 feet per second. By its use 
the computations required in investigating pipes with Table 
XVI on page 168 may be largely avoided. For example, let 
it be required to find the head lost in friction in a pipe one 
foot in diameter and 12 570 feet long when the velocity is 2.5 
feet per second. By interpolation in the table the friction-head 
is found to be 0.235 feet for 100 feet of pipe, and hence the 
total head lost in friction is 125.7 X 0.235 = 29.5 feet. 



4H 



APPENDIX. 



TABLE XXVI. COMMON LOGARITHMS. 



71 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


42 


II 


0414 


0453 


0492 


0531 
0899 


0569 


0607 


0645 


0682 


0719 


0755 


33 


12 


0792 


0828 


0864 


0934 


0969 


1004 


1038 


1072 


1 106 


35 


x 3 


"39 


"73 


1206 


1239 


1 27 1 


1303 


1335 


1367 


1399 


143° 


32 


14 


1461 


1492 


1523 


!553 


1584 


1614 


1644 


1673 


1703 


1732 


30 


J 5 


1761 


1790 


1818 


1847 


1875 


1903 


l 93 1 


J 959 


1987 


2014 


28 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


27 


17 


2304 


2330 


2355 


2380 


2405 


2430 


2 455 


2480 


2504 


2529 


25 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


24 


19 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


22 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


21 


21 


3222 


3 2 43 


3263 


3284 


3304 


3324 


3345 


3365 


3385 


3404 


20 


22 


3424 


3444 


3464 


3483 


3502 


3522 


354i 


356o 


3579 


3598 


19 


2 3 


3617 


3636 


3655 


3674 


3692 


37 11 


3729 


3747 


3766 


3784 


18 


24 


3802 


3820 


3838 


3856 


3874 


3892 


3909 


3927 


3945 


3962 


18 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


17 


26 


4150 


4166 


4183 


4200 


4216 


4232 


4249 


4265 


4281 


4298 


17 


27 


43*4 


433° 


4346 


4362 


4378 


4393 


4409 


4425 


4440 


4456 


16 


28 


4472 


4487 


4502 


4518 


4533 


4548 


45 6 4 


4579 


4594 


4609 


15 


29 


4624 


4639 


4654 


4669 


4683 


4698 


4713 


4728 


4742 


4757 


*5 


30 


477i 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


14 


31 


4914 


4928 


4942 


4955 


4969 


4983 


4997 


501 1 


5024 


5033 


14 


32 


5Q5 1 
5185 


5065 


5°79 


5092 


5io5 


5"9 


5 J 32 


5 T 45 


5*59 


5 X 72 


13 


33 


5198 


5211 


5224 


5237 


5250 


5263 


5276 


5289 


5302 


13 


34 


53i5 


5328 


534o 


5353 


5366 


5378 


539i 


5403 


54i6 


5423 


l 3 


35 


544i 


5453 


5465 


5478 


549o 


5502 


55U 


5527 


5539 


555i 


12 


36 


5563 


5575 


5537 


5599 


561 1 


5623 


5 6 35 


5 6 47 


5658 


H°. 


12 


37 


5682 


5694 


5705 


5717 


5729 


574o 


5752 


5763 


5775 


5786 


12 


33 


5793 


5809 


5821 


5832 


5843 


5855 


S866 


5377 


5SSS 


5S99 


11 


39 


59i 1 


5922 


5933 


5944 


5955 


5966 


5977 


5983 


5999 


6010 


11 


40 


6021 


6031 


6042 


6053 


6064 


6075 


60S 5 


6096 


6107 


6117 


11 


4i 


6128 


6138 


6149 


6160 


6170 


61S0 


6191 


6201 


6212 


6222 


11 


42 


6232 


6243 


6253 


6263 


6274 


62S4 


6294 


6304 


6314 


6325 


10 


43 


6335 


6345 


6355 


6365 


6375 


6385 


6395 


6405 


6415 


6425 


10 


44 


6435 


6444 


6454 


6464 


6474 


64S4 


6493 


6503 


65*3 


6522 


10 


45 


6532 


6542 


6551 


6561 


6571 


6580 


6590 


6599 


6609 


6618 


10 


46 


6628 


6637 


6646 


6656 


6665 


6675 


6684 


6693 


6702 


6712 


9 


47 


6721 


6730 


6739 


6749 


67 5 s 


6767 


6776 


6785 


6794 


6S03 


9 


43 


6812 


6821 


6830 


6S39 


6S4S 


6857 


6866 


6875 


6S84 


6S93 


9 


49 


6902 


691 1 


6920 


692S 


6937 


6946 


6955 


6964 


6972 


69S1 


9 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 


7067 


I 


5 l 


7076 


7084 


7093 


7101 


7110 


7118 


7126 


7135 


7i43 


7152 


5 2 


7160 


716S 


7177 


71S5 


7193 


7202 


7210 


7218 


7226 


7235 


S 


53 


7243 


7251 


7259 


7267 


7275 


72S4 


7292 


7300 


730S 


73 T 6 


S 


54 


7324 


•TIT* 


7340 


7343 


7356 


7364 


7372 


73So 


73SS 


7396 


s 


u 





I 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 



COMMON L GA AY THMS. 



415 



TABLE XXVI. COMMON LOGARITHMS, 



n 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


5 £ 


7404 


7412 


7419 


7427 


7435 


7443 


745 r 


7459 


7466 


7474 


8 


56 


7482 


7490 


7497 


7505 


7513 


7520 


7528 


7536 


7543 


755 1 




5 l 


7559 


7566 


7574 


7582 


7589 


7597 


7604 


7612 


7619 


7627 




58 


7634 


7642 


7649 


7657 


7664 


7672 


7679 


7686 


7694 


7701 




59 


7709 


7716 


7723 


773 1 


7738 


7745 


7752 


7760 


7767 


7774 




60 


7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7846 


7 


61 


7853 


7860 


7868 


7875 


7882 


7889 


7896 


7903 


7910 


7917 




62 


7924 


793 1 


7938 


7945 


7952 


7959 


7966 


7973 


7980 


7987 




£>z 


7993 


8000 


8007 


8014 


8021 


8028 


8035 


8041 


8048 


8055 




64 


8062 


8069 


8075 


8082 


8089 


8096 


8102 


8109 


8116 


8122 




65 


8129 


8136 


8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 


7 


66 


8i95 


8202 


8209 


8215 


8222 


8228 


8235 


8241 


8248 


8254 




67 


8261 


8267 


8274 


8280 


8287 


8293 


8299 


8306 


8312 


8319 




68 


l^ll 


8331 


8338 


8344 


835i 


8357 


8363 


8370 


8376 


8382 




69 


8388 


8395 


8401 


8407 


8414 


8420 


8426 


8432 


8439 


8445 




70 


8451 


8457 


8463 


8470 


8476 


8482 


8488 


8494 


8500 


8506 


6 


7i 


8513 


8519 


8^25 


8531 


8537 


8543 


8549 


8555 


8561 


8567 




72 


8573 


8579 


8585 


8591 


8597 


8603 


8609 


8615 


8621 


8627 




73 


8633 


8639 


8645 


8651 


8657 


8663 


8669 


8675 


8681 


8686 




74 


8692 


8698 


8704 


8710 


8716 


8722 


8727 


8733 


8739 


8745 




75 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 


6 


76 


8808 


8814 


8820 


8825 


8831 


8837 


8842 


8848 


8854 


8859 




77 


8865 


8871 


8876 


8882 


8887 


8893 


8899 


8904 


8qio 


8915 




78 


8921 


8927 


8932 
8987 


8938 


8943 


8949 


8954 


8960 


8965 


8971 




79 


8976 


8982 


8993 


8998 


9004 


9009 


9015 


9020 


9025 




80 


9031 


9036 


9042 


9047 


9053 


9058 


9063 


9069 


9074 


9079 


5 


81 


9085 


9090 


9096 


9101 


9106 


9112 


9117 


9122 


9128 


9 X 33 




82 


9138 


9 J 43 


9149 


9154 


9159 


9165 


9170 


9175 


9180 


9186 




S o 3 


9i 91. 


9196 


9201 


9206 


9212 


9217 


9222 


9227 


9232 


9238 


, 


84 


9 2 43 


9248 


9253 


9258 


9263 


9269 


9274 


9279 


9284 


9289 




85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


933o 


9335 


934o 


5 


86 


9345 


935o 


9355 


9360 


9365 


937o 


9375 


9380 


9385 


9390 




87 


9395 


9400 


9405 


9410 


9415 


9420 


9425 


9430 


9435 


9440 




88 


9445 


945o 


9455 


9460 


9465 


9469 


9474 


9479 


9484 


9489 




89 


9494 


9499 


95°4 


9509 


9513 


9518 


9523 


9528 


9533 


9538 




90 


9542 


9547 


9552 


9557 


9562 


9566 


957i 


9576 


9581 


9586 


5 


9 1 


959o 


9595 


9600 


9605 


9609 


9614 


9619 


9624 


9628 


9633 




92 


9638 


9643 


9647 


9652 


9657 


9661 


9666 


9671 


9675 


9680 




93 


9685 


9689 


9694 


9699 


9703 


9708 


9713 


9717 


9722 


9727 




94 


973i 


9736 


974i 


9745 


9750 


9754 


9759 


9763 


9768 


9773 




95 


9777 


9782 


9786 


9791 


9795 


9800 


9805 


9809 


9814 


9818 


4 


96 


9823 


9827 


9832 


9836 


9841 


9845 


9850 


9854 


9859 


9863 




97 


9868 


9872 


9877 


9881 


98S6 


9890 


9894 


9899 


9903 


9908 




98 


9912 


9917 


9921 


9926 


993o 


9934 


9939 


9943 


9948 


9952 




99 


9956 


9961 


9965 


9969 


9974 


9978 


9983 


9987 


9991 


9996 




n 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 



416 



APPEXDIX. 



TABLE XXVII. SQUARES OF NUMBERS. 



n. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


1.0 


1. 000 


1.020 


1.040 


1. 061 


1.082 


1-103 


1. 124 


I-I45 


1.166 


1. 188 


22 


I.I 


1.210 


1.232 


1.254 
1.488 


1.277 


1.300 


1-323 


1.346 


1.369 


1.392 


1.416 


24 


1.2 


1.440 


1.464 


T -5 T 3 


1-538 


J-563 


1.588 


1-613 


1.638 


1.664 


26 


i-3 


1.690 


1. 716 


1.742 


1.769 


1.796 


1.823 


1.850 


1.877 


1.904 


1.932 


28 


1.4 


1.960 


1.988 


2.016 


2.045 


2.074 


2.103 


2.132 


2. 161 


2.190 


2.220 


30 


i-5 


2.250 


2.280 


2.310 


2.341 


2.372 


2.403 


2-434 


2.465 


2.496 


2.528 
2.856 


32 


i.6 


2.560 


2.592 


•2.624 


2.657 


2.690 


2.723 


2.756 


2.789 


2.822 


34 


i-7 


2.890 


2.924 


2.958 


2-993 


3.028 


3-063 


3.098 


3- x 33 


3.168 


3.204 


% 


i.8 


3.240 


3.276 


3-3 12 


3-349 


3-386 


3423 


3.460 


3-497 


3-534 


3-572 


38 


1.9 


3.610 


3.648 


3.686 


3-725 


3-764 


3-803 


3.842 


3.881 


3.920 


3.960 


40 


2.0 


4.000 


4.040 


4.080 


4.121 


4.162 


4.203 


4.244 


4.285 


4.326 


4.368 


42 


2.1 


4.410 


445 2 


4.494 


4-537 


4.580 


4.623 


4.666 


4709 


4-752 


4.796 


44 


2.2 


4.840 


4.884 


4.928 


4-973 


5.018 


5-063 


5.108 


5-153 


5 if 


5.244 


46 


2-3 


5.290 


5-336 


S 'fl 


5429 


5476 


5-523 


5-57o 


5-6i7 


5.664 


5-712 


48 


2.4 


5.760 


5.808 


5.856 


5-905 


5-954 


6.003 


6.052 


6.101 


6.150 


6.200 


50 


2-5 


6.250 


6.300 


6.350 
6.864 


6.401 


6.452 


6.503 


6-554 


6.605 


6.656 
7.182 


6.708 


52 


2.6 


6.760 


6.812 


6.917 


6.970 


7.023 


7.076 


7.129 


7.236 


54 


2.7 


7.290 


7-344 


7-398 


7453 


7.508 


7.563 


7.618 


7-673 


7.728 


7.784 


56 


2.8 


7.840 


7.896 


7.952 


8.009 


8.066 


8.123 


8.180 


8.237 


8.294 


8.352 


58 


2.9 


8.410 


8.468 


8.526 


8.585 


8.644 


8.703 


8.762 


8.821 


8.880 


8.940 


60 


3-o 


9.000 


9.060 


9.120 


9.181 


9.242 


9-303 


9-364 


9425 


9.486 


9-548 


62 


3-i 


9.610 


9.672 


9-734 


9-797 


9.860 


9-923 


9.986 


10.05 


10.11 


10.18 


6 


3- 2 


10.24 


10.30 


10.37 


10.43 


10.50 


10.56 


10.63 


10.69 


10.76 


10.82 


7 


3-3 


10.89 


10.96 


11.02 


11.09 


11. 16 


11.22 


11.29 


11.36 


11.42 


11.49 


7 


34 


11.56 


11.63 


11.70 


11.76 


11.83 


11.90 


11.97 


12.04 


12. II 


12.18 


7 


3-5 


12.25 


12.32 


12.39 


12.46 


12.53 


12.60 


12.67 


12.74 


12.82 


12.89 


7 


3-6 


12.96 


i3-°3 


13.10 


13.18 


!3- 2 5 


l 3-3 2 


13.40 


1347 


13.54 


13.62 


7 


H 


13.69 


I3-76 


13.84 


i3-9i 


13-99 


14.06 


14.14 


14.21 


I4.29 


14-36 


8 


3-8 


14.44 


14.52 


14-59 


14.67 


14-75 


14.82 


14.90 


14.98 


15-05 


15.13 


8 


3-9 


15.21 


15.29 


x 5-37 


15-44 


r 5-5 2 


15.60 


15.68 


15.76 


I5.84 


I5-92 


8 


4.0 


16.00 


16.08 


16.16 


16.24 


16.32 


16.40 


16.48 


16.56 


16.65 


16.73 


8 


4.1 


16.81 


16.89 


16.97 


17.06 


17.14 


17.22 


I7-3 1 


17-39 


1747 


17.56 


8 


4.2 


17.64 


17.72 


17.81 


17.89 


17.98 


18.06 


18.15 


1S.23 


lS.32 


18.40 


9 


4-3 


18.49 


18.58 


18.66 


18.75 


18.84 


18.92 


19.01 


19.10 


I9.18 


19.27 


9 


44 


19.36 


1945 


J 9-54 


19.62 


19.71 


19.80 


19.S9 


19.98 


20.07 


20.16 


9 


4-5 


20.25 


20.34 


20.43 


20.52 


20.61 


20.70 


20.79 


20.SS 


2O.9S 


21.07 


9 


4.6 


21.16 


21.25 
22.18 


21.34 


21.44 


21-53 


21.62 


21.72 


21.81 


2I.9O 


22.00 


9 


4-7 


22.09 


22. 2S 


22.37 


22.47 


22.56 


22.66 


22.75 


22.85 


22.94 


10 


4.8 


23.04 


23.14 


23-23 


2 3-33 


23-43 


2 3-5 2 


23.62 


23-72 


23.81 


23-91 


10 


4.9 


24.01 


24.11 


24.21 


24.30 


24.40 


24-5° 


24.60 


24.70 


24.S0 


24.90 


10 


5-o 


25.00 


25.10 


25.20 


25-30 


25.40 


25-50 


25.60 


25.70 


2;.$ I 


25-91 


10 


5-i 


26.01 


26.11 


26.21 


26.32 


26.42 


26.52 


26.63 


26.73 


26.83 
27.88 


26.94 


10 


5-2 


27.04 


27.14 


27.25 


27-35 


27.46 


27.56 


27-67 


27.77 


27.98 


11 


5-3 


28.09 


28.20 


28.30 


28.41 


2S.52 


28.62 


2S.73 


2S.S4 


2S.94 


29.05 


11 


54 


29.16 


29.27 


29.38 


29.48 


29-59 


29.70 


29.S1 


29.92 


30.03 


30.14 


11 


n. 





1 


2 


3 


4 


5 


6 


7 


8 . 


9 


Diff. 



SQUARES OF NUMBERS. 



417 





TABLE XXVII. 


SQUARES 


OF NUMBERS. 






n. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 


H 


3°- 2 5 3°-3 6 


3°47 


30.58 


30.69 


30.80 


30.91 


31.02 


31-14 


3 J - 2 5 


11 


5.6 


31.36 31.47 


3I-53 


31.70 


31.81 


31.92 


32.04 


32.15 


32.26 


32-38 


11 


H 


32.49 32.60 


32.72 


32.83 


32-95 


33-o6 


33-i8 


33-29 


3341 


33-52 


12 


5-3 


33-64 3376 


33-87 


33-99 


34-11 


34.22 


34-34 


3446 


34-57 


34-69 


12 


5-9 


34.81 34.93 


35-o5 


35-i6 


35-28 


35-4Q 


35-52 


35.64 35.76 


35.88 


12 


6.0 


36.00 36.12 


36.24 


36.36 


36.48 


36.60 


36.72 


36.84 


36.97 


37-09 


12 


6.1 


37-2i 37 . 33 


3745 


37-58 


37-70 


37-82 


37-95 


38.07 


38.19 


38.32 


12 


6.2 


38.44 38.56 


38.69 


38.81 


38.94 


39.06 


39-19 


39-3 1 


39-44 


39-56 


13 


6-3 


39.69 39.82 


39-94 


40.07 


40.20 


40.32 


4045 


40.58 


40.70 


40.83 


13 


6.4 


40.96 41.09 


41.22 


41-34 


41.47 


41.60 


41-73 


41.86 


41.99 


42.12 


J 3 


6.5 


42.25 42.38 


42.51 


42.64 


42.77 


42.90* 


43-03 


43.16 


43-30 


43-43 


13 


6.6 


43-56 43-69 


43.82 


43-96 


44.09 


44.22 


44-36 


44.49 


44.62 


44.76 


13 


6. 7 


44.89 45.02 


45.16 


45-29 


45-43 


45-56 


457o 


45-83 


45-97 


46.10 


14 


6.8 


46.24 46.38 


46.51 


46.65 


46.79 


46.92 


47.06 


47.20 


47-33 


47-47 


14 


6.9 


47-6i 47-75 


47.89 


48.02 


48.16 


48.30 


48.44 


48.58 


48.72 


48.86 


14 


7.0 


49.00 49.14 


49.28 


49.42 


49-56 


49.70 


49.84 


49.98 


50-13 


50.27 


14 


7-i 


50.41 5o-55 


50.69 


50.84 


50.98 


51.12 


5*- 2 7 


5J-4I 


51-55 


5!-7o 


14 


7.2 


51.84 51.98 


5 2 -J3 


52.27 


s? "il 


52-56 


5 2 7i 


52.85 


53-oo 


53- J 4 


15 


7-3 


53.29 53.44 


53-58 


53-73 


53.88 


54.02 


54-17 


54-32 
55.80 


54-46 


54.61 


15 


74 


54.76 54.91 


55-o6 


55-20 


55-35 


55-5o 


55-65 


55-95 


56.10 


15 


7-5 


56.25 56.40 


56-55 


56.70 


56.85 


17.00 


57-15 


57.30 


57-46 


57-6i 


15 


7.6 


57.76 57.91 


58.06 


58.22 


58.37 


58.52 


58.68 


58.83 


58.98 


59-14 


15 


77 


59.29 59.44 


59.60 


59-75 


59-91 


60.06 


60.22 


60.37 


60.53 


60.68 


16 


7.8 


60.84 61.00 


61.15 


61.31 


61.47 


61.62 


61.78 


61.94 


62.09 


62.25 


16 


7-9 


62.41 62.57 


62.73 


62.88 


63.04 


63.20 


^3-3^ 


63-52 


63.68 


63.84 


16 


8.0 


64.00 64.16 


64.32 


64.48 


64.64 


64.80 


64.96 


65.12 


65.29 


6545 


16 


8.1 


65.61 65.77 


65-93 


66.10 


66.26 


66.42 


66.59 


66.75 


66.91 


67.08 


16 


8.2 


67.24 67.40 


67-57 


67.73 


67.90 


68.06 


68.23 


68.39 


68.56 


68.72 


17 


I' 3 


68.89 69.06 


69.22 


69-39 


69.56 


69.72 


69.89 


70.06 


70.22 


70.39 


17 


8.4 


70.56 70.73 


70.90 


71.06 


71.23 


71.40 


71-57 


71-74 


71.91 


72.08 


17 


8.5 


72.25 72.42 


72.59 


72.76 


72.93 


73.10 


73-27 


73-44 


73.62 


73-79 


17 


8.6 


73-96 74-13 


74-3° 


74.48 


74.65 


74.82 


75.00 


75- J 7 


75-34 


75-52 


17 


8.7 


75.69 75.86 


76.04 


76.21 


76.39 


76.56 


76.74 


76.91 


77.09 


77.26 


18 


8.8 


77.44 77.62 


77-79 


77-97 


78.15 


78.32 


78.50 


78.68 


78.85 


79-03 


18 


8.9 


79.21 79.39 


79-57 


79-74 


79.92 


80.10 


80.28 


80.46 


80.64 


80.82 


18 


9.0 


81.00 81.18 


81.36 


81.54 


81.72 


81.90 


82.08 


82.26 


82.45 


82.63 


18 


9.1 


82.81 82.99 


83.17 


83-36 


83-54 


83.72 


83.91 


84.09 


84.27 


84.46 


18 


9.2 


84.64 84.82 


85.01 


85.19 


85.38 


85.56 85.75 


85-93 


86.12 


86.30 


19 


9-3 


86.49 86.68 


86.86 


87.05 


87.24 


87.42 


87.61 


87.80 


87.98 


88.17 


19 


94 


88.36 88.55 


88.74 


88.92 


89.11 


89.30 


89.49 


89.68 


89.87 


90.06 


J 9 


9-5 


90.25 90.44 


90.63 


90.82 


91.01 


91.20 


9 l -39 


91.58 


91.78 


91.97 


19 


9.6 


92.16 92.35 


92-54 


92.74 


92-93 


93.12 


93-32 


93-51 


93-70 


93-90 


19 


9-7 


94.09 94.28 


94.48 


94.67 


94.87 


95.06 


95.26 


9545 


95-65 


95-84 


20 


9.8 


96.04 96.24 


9643 


96.63 


96.83 


97.02 


97.22 


97.42 


97.61 


97-81 


20 


9.9 


98.01 98.21 


98.41 


98.60 


98.80 


99.00 


99.20 


99.40 


99.60 


99.80 


20 


n. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


Diff. 



4i8 



APPEXDIX. 



TABLE XXVIII. AREAS OF CIRCLES. 



d 





1 2 3 


4 


5 


6 


7 


8 9 


iDiff. 


I.O 


■ 7854 


.S012 .8171 .S332 


•S495 


.8659 


.8825 


.8992 


.9161 .9331 




I.I 


.9503 


.9677 .9S52 1.003 


1.021 


1.039 


1.057 


1.075 


1.094 1. 112 




1.2 


1.131 


1. 150 1. 169 1.1SS 


I.203 


1.227 


1.247 


1.267 


1.287 1-307 


19 


i-3 


1.327 


1.348 1.363 1.389 


1. 410 


I-43I 


1-453 


1.474 


1.496 1. 517 


21 


1.4 


1-539 


1. 561 1.584 1.606 


1.629 


1. 651 


1.674 


1.697 


1.720 1.744 


22 


i-5 


1.767 


1. 791 1-815 1.839 


1.863 


1.887 


1. 911 


1.936 


1. 961 1.986 


24 


1.6 


2. on 


2.036 2.061 2.087 


2. 112 


2.133 


2.164 


2.190 


2.217 2.243 


26 


1-7 


2.270 


2.297 2.324 2.351 


2.373 


2.405 


2-433 


2.461 


2.488 2.516 


27 


1.3 


2-545 


2.573 2.6C2 2.63O 


2.659 


2.68S 


2.717 


2.746 


2.776 2.806 


29 


1.9 


2.S35 


2.S65 2.S95 2.926 


2.956 


2.986 


3-oi7 


3.048 


3.079 3. no 


30 


2.0 


3-142 


3.173 3-205 3-237 


3.269 


3-301 


3-333 


3-365 


3.398 3.431 


32 


2.1 


3-464 


3-497 3-530 3.563 


3-597 


3.631 


3.664 


3.698 


3-733 3-767 


34 


2.2 


3.801 


3.S36 3.871 3.906 


3-941 


3.976 


4.012 


4.047 


4.083 4.119 


35 


2.3 


4-155 


4.191 4.227 4.264 


4.301 


4-337 


4-374 


4.412 


4-449 4-4S6 


36 


2.4 


4-524 


4.562 4.600 4.63S 


4.676 


4.714 4-753 


4.792 


4.S31 4.S70 


33 


2-5 


4.909 


4.94S 4.93S 5.027 


5.067 


5.107 


5.147 


5.1S7 


5.228 5.269 


40 


2.6 


5-309 


5.350 5-39 1 5-433 


5-474 


5-515 


5-557 


5-599 


5.641 5.683 


4i 


2.7 


5.726 


5-763 5.811 5.853 


5.896 


5.940 


5-9S3 


6.026 


6.070 6. 1 14 


43 


2.8 


6.158 


6.202 6.246 6.290 


6-335 


6.379 


6.424 


6.469 


6.514 6.560 


44 


2.9 


6.605 


6.651 6.697 6.743 


6.7S9 


6.835 


6.S81 


6.928 


6.975 7.022 


46 


3-0 


7.069 7. 116 7.163 7. 211 


7-25S 


7.306 


7-354 


7.402 


7.451 7.499 


4S 


3-i 


7.548 7-596 7-645 7.694 


7-744 


7-793 


7-S43 


7.S92 


7.942 7-992 


49 


3-2 


8.042 


8.093 8.143 8.194 


8.245 


S.296 


8-347 


8.39S 


S.450 8.5 : 


51 


3-3 


8-553 


8.605 8.657 8.709 


8.762 


S.S14 


8.S67 


S.920 


3. 973 9.026 


52 


3-4 


9.079 9.133 9.1S6 9.240 9.294 


9-343 


9.402 9.457 9.511 9.566 


54 


3-5 


9.621 


9.676 9.731 9.787 9.S42 


9.89S 


9-954 


10.01 


10.07 10.12 


56 


3-6 


10.18 


10.24 10.29 10.35 


10.41 


10.46 


10.52 


10.5S 


10.64 10.69 


6 


3-7 


10.75 


10. Si 10.S7 10.93 


10. eg 


11.04 


n. 10 


11. 16 


11.22 11.28 


6 


3-S 


11.34 


11.40 11.46 11.52 


11.58 


11.64 


11.70 


11.76 


11.82 11. S3 


6 


3-9 


H-95 


12.01 12.07 12.13 


12.19 


12.25 


12.32 


12.3S 


12.44 12.50 


6 


4.0 


12.57 


12.63 12.69 12.76 


12. 82 


12. S3 


12.95 


13.01 


13.07 13.14 


7 


4.1 


13.20 


13.27 13 33 13.40 


I3.46 


13-53 


13-59 


13.66 


13.72 13.79 


7 


4.2 


13.85 


13.92 13.99 J 4-05 


14.12 


14.19 


14.25 


14.32 


14.39 14-45 


7 


4-3 


14-52 


14.59 14-66 14.73 


14.79 


14.S6 


14-93 


15.00 


15-07 15-14 


7 


4-4 


15.21 


15.27 15.34 15.41 


15.48 


15-55 


15 62 


15.69 


15.76 15.53 


" 


4-=; 


15.90 


15. 9S 16.05 16.12 


16.19 


16.26 


16.33 


16.40 


16.47 16.55 


7 


4.6 


16.62 


16.69 16.76 16. S4 


16.91 


16.9S 


17.06 


17-13 


17.20 17.18 


7 


4-7 


17-35 


17.42 17.50 17.57 


I7.65 


17.72 


17. So 


17.87 


17.95 1S.02 


a 


4.8 


1S.10 


18.17 18.25 18.32 


1S.40 


18.47 


15.55 


1S.63 


1S.70 1S.7S 


s 


4.9 


18.86 


18.93 19.01 19.09 


19.17 


19.24 


19.32 


19.40 


19.4S 19.56 


s 


5-0 


19.63 


19.71 19.79 19.S7 


19-95 


20.03 


20.11 


20.19 


20.27 20.35 


5 


5-1 


20.43 


20.51 20.59 20.67 


20.75 


20. S3 


20.91 


20.99 


21.07 21.16 


3 


5-2 


21.24 


21.32 21.40 21.48 


21-57 


21.65 


21-73 


21. Si 


21.90 21. 9S 


S 


5o 


22.06 


22.15 22.23 22.31 


22.40 


22. 4S 


22.56 


22.65 


22.73 22. 32 


S 


5-4 


22.90 


22.99 23.07 23.16 


23.24 


23.33 


23-41 


23.50 


23.59 23.67 


9 


</ 





1 2 3 


4 


5 


6 


7 


S 9 


Diff. 



AREAS OF CIRCLES. 
TABLE XXVIII. AREAS OF CIRCLES.— Continued. 



419 



d 





1 


2 3 4 


5 6 


7 


8 


9 


Diff. 


5.5 


23.76 


23.S4 


23.93 24.02 24.11 


24.19 24.28 


24-37 


24.45 


24-54 


9 


5 


6 


24.63 


24.72 


24. Si 24. 89 24.98 


25.07 25.16 


25.25 


25-34 


25-43 


9 


5 


7 


25-52 


25.61 


25.70 25.79 25-88 


25.97 26.06 


26.15 


26.24 


26.33 


9 


5 


8 


26.42 


26.51 


26.60 26.69 26.79 


26.88 26.97 


27.06 


27-15 


27.25 


9 


5 


9 


27-34 


27-43 


27.53 27.62 27.71 


27. Si 27.90 


27.99 


28. 09 


27.18 


9 


6 





28.27 


28.37 


28.46 28.56 28.65 


2S.75 28.84 


28.94 


29.03 


29.13 


9 


6 


1 


29.22 


29.32 


29.42 29.51 29.61 


29.71 29.80 


29.90 


30.00 


30.09 


10 


6 


2 


30.19 


30.29 


30.39 30.48 30.58 


30.68 30.78 


30.88 


30.97 


3I-07 


10 


6 


3 


31.17 


31.27 


31.37 31-47 31-57 


31-67 31-77 


31.87 


31-97 


32.07 


10 


6 


4 


32.17 


32.27 


32.37 32.47 32.57 


32.67 32.78 


32.88 


32.98 


33-o8 


10 


6 


5 


33-i3 


33.29 


33-39 33-49 33 59 


33.70 33.80 


33.90 


34 00 


34-11 


10 


6 


6 


34.21 


34.32 


34.42 34.52 34.63 


34-73 34.84 


34-94 


35.05 


35.15 


TO 


6 


7 


35-26 


35-36 


35-47 35-57 35-6S 


35.78 35.89 


36.00 


36.10 


36.21 


IO 


6. 


S 


36.32 


36.42 


36.53 36.64 36.75 


36.85 36.96 


37 07 


37.18 


37.28 


II 


6 


9 


37-39 


37.50 37.61 37.72 37.83 


37.94 38.05 


33.16 


3S.26 


38.37 


II 


7- 





38.48 33.59 


38.70 38.82 38.93 


39.04 39.15 


39.26 


39-37 


39-48 


II 


7 


1 


39-59 


39-7o 


39.82 39.93 40.04 


40.15 40.26 


40.38 


40.49 


40.60 


II 


7 


2 


40.72 


40.83 


40.94 41.06 41.17 


41.28 41.40 


4I.5I 


41.62 


41.74 


II 


7 


3 


41.85 


41.97 42.08 42.20 42.31 


42.43 42.54 


42.66 


42.78 42.89 


II 


7 


4 


43.01 


43-12 


43.24 43.36 43.47 


43-59 43.71 


43.83 


43.94 44-o6 


12 


7 


5 


44.18 


44oO 


44-41 44.53 44.65 


44-77 44-89 


45.01 


45.13 


45.25 


12 


7 


6 


45.36 


45.48 


45.60 45.72 45.84 


45.96 46.08 46.20 


46.32 


46.45 


12 


7 


7 


46.57 


46.69 46.81 46.93 47.05 


47-17 47-29 47-42 47.54 47-66 


12 


7- 


8 


47.78 47.91 


48. 03 48.15 48.27 


48.40 48.52 


48.65 


48.77 


4S.89 


12 


7. 


9 


49.02 


49.14 


49.27 49.39 49.51 


49-64 49-76 49-89 


50.01 


50.14 


12 


8. 





50.27 


50.39 


50.52 50.64 50.77 


50.90 51.02 


5I.I5 


51. 28 


51.40 


13 


8 


1 


51.53 


51.66 


51.78 51.91 52.04 


52.17 52.30 


52.42 


52.55 


52.68 


13 


8 


2 


52.81 


52.94 


53.07 53-20 53.33 


53-46 53-59 


53.72 


53.85 


53 98 


13 


8 


3 


54-11 


54-24 


54-37 54 5o 54.63 


54.76 54.89 


55.02 


55.15 


55-29 


13 


8 


4 


55-42 


55-55 


55.68 55.81 55.95 


56.08 56.21 


56.35 


56.48 


56.61 


13 


8 


5 


56.75 56.S8 


57-OI 57.15 57. 2S 


57.41 57.55 


57.68 


57-32 


57-95 


13 


8 


6 


58.09 


58.22 


5S.36 5S.49 5S.63 


58.77 58.90 


59-04 


59-17 


59 3i 


14 


8 


7 


59-45 


59-58 


59.72 59.86 59-99 


60.13 60.27 


60.41 


60.55 


60.68 


14 


8 


8 


60.82 


60.96 


61.10 61.24 61.38 


61.51 61.65 


61.79 


6i.93 


62.07 


14 


8 


9 


62.21 


62.35 


62.49 62.63 62.77 


62.91 63.05 


63.19 


63-33 


63-48 


14 


9 





63.62 


63.76 63.90 64.04 64.18 


64.33 64.47 


64.61 64.75 


64.90 


14 


9 


1 


65.04 


65.18 


65-33 65.47 65.61 


65.76 65.90 


66.04 


66.19 


66.33 


14 


9 


2 


66.48 


66 62 


66.77 66.91 67.06 


67.20 67.35 67.49 


67.64 


67. 7S 


15 


9 


3 


67.93 


6S.08 


68.22 6S.37 6S.51 


63.66 68.81 


63.g6 69.10 


69.25 


15 


9 


4 


69.40 69.55 


69.69 69.84 69.99 


70.14 70.29 


70.44 


70.58 


7073 


15 


9 


5 


70:88 


71-03 


71. iS 71.33 71.48 


71.63 71. 78 


71.93 


72.08 


72.23 


15 


9 


6 


72.38 


72.53 


72.68 72.84 72.99 


73 14 73-29 


73-44 


73-59 


73-75 


15 


9 


7 


73-90 


74-05 


74.20 74.36 74-51 


74.66 74. 82 


74-97 


75-12 


75-28 


15 


9 


8 


75-43 


75.58 


75.74 75.89 76.05 


76.20 76.36 


76.51 


76.67 76.82 


16 


9 


9 


76.98 77.13 


77.29 77.44 77.60 


77.76 77.91 


73.o7 


78.23 


78.38 


16 


d 





1 


2 3 4 


5 6 


7 


8 


9 


Diff. 



420 



APPENDIX. 



TABLE XXIX, LOSS OF HEAD IN FRICTION 
FOR 100 FEET OF PIPE. 





Diameter 

in 

Feet. 


1 




Velocity in Feet per Second. 








1 


2 


3 


4 


6 


10 


15 






Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 


Feet. 




O.05 


I.46 


5.IO 


10.3 


16.9 


34-7 








O.I 


0-59 


1.99 


4.20 


6.97 


14-5 


37-3 






O.25 


.20 


0. 70 


I.46 


2.-10 


5 -37 


13-7 


29.4 




0-5 


.09 


0.32 


O.70 


1. 14 


2.46 


O. 22 


13-3 




0.75 


•05 


.21 


•45 


73 


T-57 


3-94 


8.40 




I. 


.04 


•15 


• 32 


• 55 


1. 12 


2. So 


5-95 




1-25 


•03 


. II 


.25 


.42 


0.S5 


2. 11 


4.48 




i-5 


.02 


.09 


.20 


•33 


.67 


1.66 


3-50 




i-75 


.02 


.07 


.16 


.26 


•54 


i-33 


2.80 




2. 


.02 


.06 


•13 


.21 


•45 


1.09 


2.27 




2.5 


.01 


•05 


.10 


.16 


•34 


0.S1 


1.68 




3- 


.01 


.04 


.07 


.12 


.26 


.67 


1.40 




3-5 


.OI 


•03 


.06 


. 10 


.21 


•53 






4- 


.01 


.02 


• 05 


.08 


•17 


.42 






5- 


.00 


.02 


.04 


.06 


•13 








6. 


.00 


.OI 


•03 


.05 


.10 







INDEX. 



421 



INDEX. 



Abbot, H. L., 265 
Absolute velocity, 68, 319, 337, 400 
Acceleration, 3, 10, n, 70 
Adjutage, 137 

Advantageous section, 225, 228, 240 
velocity, 325, 356, 387 
Advice to students, 13 
Air pressure, 7, 17, 395 
Anchor ice, 4 
Angular velocity, 59 
Answers to problems, 410 
Approach, angle of, 367 

velocity of, 46, 105, 133 
Archimedes, 30 
Areas of circles, 418 
Atmospheric pressure, 7, 8, 17, 49, 146 

Backpitch wheel, 371 
Backwater, 277 

function, 280 
Barometer, 7, 396 
Bazin, H., 223, 226, 263, 351 
Bernouillt, D., 137 
Bidone, G., 88, 277, 307 
Bodmer, G. R., 382 
Boiling points, 8 
Borda, J. C, 73 
Bossut, C., 73, 75, 128 
Bowie, A. ]., 85, 86, 363 
Boyden, U., 100, 400 
Boyden diffuser, 400 

turbine, 300, 384 
Breckenridge, L. P., 303 
Bresse, M., 279 



Breast wheels, 358, 371 
Brick conduits, 231 

sewers, 237 
Brooks, 247 
Browne, R. E., 363 
Buckets, 301, 332, 355 
Buff, H., 147 



402 



Canals, 212-246, 
Canal lock, 96 
Cascade wheel, 363 
Castel, 128, 130 
Centre of buoyancy, 34 
of gravity, 18, 34 
of pressure, 22, 24 
Centrifugal force, 58, 329, 368 
Channels, 212, 250, 313 
Church, I. P., 365 
Cippoletti, 125, 126 
Circles, areas of, 413, 418 
properties of, 222 
Circular conduits, 218 

cross-sections, 222 
orifices, 45, 78 
Classification of surfaces, 233 
of turbines, 376 
Coal used by steamers, 340 
Coefficient of contraction, 73, 78 
in nozzles, 133 
in tubes, 131, 152 
Coefficient of discharge, 76 

in channels, 217, 229, 239 
in conduits, 218, 227 
in compound tubes, 137 



4-2 



INDEX. 



Coefficient of discharge, 76 
in orifices, 79-92 
in pipes, 163 
in sewers, 238 
in tubes, 128-142 
in turbines, 384 
in weirs, 108, in, 125 

Coefficient of roughness, 233 

Coefficient of velocity, 74 

in tubes, 128, 131, 152 

Compound pipes, 184 

tubes, 137, 158 

Compressibility of water, 9 

Computations, 12, 413 

Conduit pipe, 188 

Conduits, 212-246 

Cone nozzles, 134 

Conical tubes, 130 
wheel, 372 

Conservation of energy, 38 

Contracted weirs, 108 

Contraction of a jet, 72, 73 
coefficient of, 73 
gradual, 152 
sudden, 151 
suppression of, 88 

Cotton hose, 207 

Coxe, E. B., 309. 

Crest of a weir, 99 

rounded and wide, 117 

Critical velocity, 210 

Cubic foot, 5 

Current indicators, 25S 
meters, 258, 267 

Curve of backwater, 278 

Curved surfaces, 21 

Curves in pipes, 170, 313 

Darcy, H., 223, 226, 252 
Dams, 19, 28, 31, 119, 121, 278 
Danalde, 372 
D'Aubissox, J. F., 130 
Depth of flotation, 31 
Design of turbines, 389 

of water wheels, 352, 354 



Diameters of pipes, 177 

water mains, 202 
Diffuser, 400 
Discharge, 37, 94 

coefficient of, 76 

conduits, 212-246 

energy of, 92 

orifices, 71-97 

tubes, 128-158 

pipes, 162-211 

rivers, 247-282 

turbines, 382 
Discharging capacity, 182 
Disk valve, 171 
Distilled water, 6 
Ditches, 227, 239 
Diverging tubes, 137 
Downing, S., 94 
Downward-flow wheels, 369, 372 

turbines, 376, 394 
Dropping head, 94 
Du Bois, A. J., 372 

DUBUAT, N., 242 

Dyer, C. W. D., 126 
Dynamic pressure, 67, 304-318 
Dynamometer, 2S2 

Effective head, 93, 142, 163 
power, 2S3, 290 

Efficiency, 65, 284, 353 
of jet, 93 

of jet propeller, 342 
of motors, 294 
of moving vanes, 323 
of paddle wheels, 345 
of reaction wheel, 374 
of screw propeller, 346 
of turbines, 290-301, 373- 

409 
of water-wheels, 353-372 

Egg-shaped sewers, 236 

Elevations by barometer, 8 

Ellis, T. G., 82 

Emerson, J., 100 

Emptying a canal lock, 96 



INDEX. 



423 



Emptying a vessel, 56 

Energy, 2, 3S, 2S3, 352 
in channels, 245 
in tubes, 129, 139, 145 
of a jet, 64, 92, 129 

Enlargement of section, 148, 150 

Entrance angle, 367 

Equilibrium, 33, 34 

Errors in computations, 174 

in measurements, 91, 164, 248 

Eureka turbine, 295, 381 

Eytelwein, J. A., 73, 137, 180 

Exit angle, 367 

Faesch and Picard, 402 

Falling bodies, 10 

Fanning, J. T., 83, 168, 198 

Fire hose, 207 
service, 200 

Floats, 256, 267 

Flotation, depth of, 31 

stability of, 33 

Flow, dynamic pressure of, 303-334 
from orifices, 37, 71-97 
in canals and conduits, 212-246 
in rivers, 247-282 
through pipes, 162-211 
through tubes, 128 
through turbines, 295, 382, 406 
under pressure, 49, 395 

Flynn, P., 126, 236 

Foote, A. D., 86 

Foss, W. E., 244 

Fourneyron turbine, 377 

Francis, J. B., 4, 107, no, 112, 113, 
114, 120, 125, 126, 138, 257, 298, 
385 

Francis turbine, 377 

Freeman, J. R., 132, 134, 136, 207 

Free surface, 15 

Frictional resistances, 37 
in channels, 216 
in pipes, 166, 199, 2ir, 338 
in water wheels, 325, 367 
in turbines, 374, 391 



Frictional resistance of ships, 336 

Friction brake, 291 
factors, 168 
heads, 167, 413, 420 

Fteley, A., 107, 114, 116, 117, 118, 
231, 245 

Gallon, 1, 5 
Ganguillet, E., 233 
Gate of a turbine, 376, 379 
Gauging the flow, 71, 90, 98, 188 

of rivers, 101, 248, 266 
Girard, P., 400 
Grassi, G., 9 
Gravity, acceleration of, 10 

centre of, 18, 34 
Guides, 379; 392, 407 

Head, 15, 288 

and pressure, 15, 17 

losses of, 142, 154, 164, 166, 

242, 289 
measurement of, 91, 100, 288 
Height of jets, 129, 135 
Herschel, C, 114, 158, 160, 253, 

300, 409 
Holyoke tests, 298, 389 
Hooke gauge, 4, 100, 213, 286 
Horizontal impulse wheels, 366 

range of a jet, 61, 136 
Horse-power, 205, 283, 293, 364 
Horse-shoe conduits, 231 
Hose, 207 

House service pipes, 196 
Humphreys, A. A., 265 
Hunt turbine, 380 
Hurdy-gurdy wheel, 363 
Hydraulic gradient, 189 
jump, 275 

motors, 65, 292, 352-420 
radius, 212-226 
Hydrometric balance, 259 

pendulum, 259 
Hydrostatic head, 17, 53, 67 
Hydrostatics, 14-35 



424 



INDEX. 



Ice, 4 

Immersed bodies, 30, 316 
Impact, 306, 322, 354 
Impulse, 66, 304-327 

turbines, 377, 397 

wheels, 363-372 
Inward-flow turbines, 376, 380 
Inward projecting tubes, 141 

Jet propeller, 341 
Jets, 64, 71-76 

contraction of, 72 * 

energy of, 64, 129 

height of, 129, 135 

impulse of, 66, 304 

on vanes, 310, 319 

path of, 60 

range of, 62, 136 
Jonval turbine, 377, 395 
Jump, 276 

Knight, E. H., 345 
Knot, 335 

K utter, W. R., 229 
Kutter's formula, 229, 233, 234, 238, 
239. 247 

Lampe's formula, 20S, 232 
Landreth, O. H., 3 
Leakage, 2S7, 358, 391 
Leffel turbine, 379, 381 
Lesbros, J. A., 88, 107 
Linen hose, 207 
Lighthouses, 318 
Log, 257, 335 
Logarithms, 3, 12, 414 
Long pipes, 1S1 

Loss of head, 93, 142, 14S, 161, 163, 
170, 242 

in friction, 166. 413, 420 

measurement of, 155 
Lowell hydraulic experiments, 13S, 257 
tests, 29S 

Masonry dams, 2S 



Maximum efficiency, 354 

velocity in river, 263 
Mean velocity, 37, 214, 247, 261 
Measurement of power, 2S3-303 

of water, 72, 90, 130, 
158, 285 
Meissner, G., 382 
Mercury, 8, 52, 94 
Merriman, M., i, 9 
Metacentre, 34 
Meters, current, 254 

water, 287 

Venturi, 158 
Metric system, 2, 6 
Michelotti, F. D., 73, 76 
Mill power, 302 
Miner's inch, S4 
Module, 85 

Moments of inertia, 24, 26 
Morosi, J., 308 
Morris, I. P., i Co = , 409 
Mortar lining, 223 
Motors, 295, 352-412 
Mouthpiece, 137 
Moving vanes, 31S 

Naval architecture, 35 

hydromechanics, 335-351 
Negative pressure, 55, 146, 159, 191 
Newton. I., 73 
Niagara turbines, 376, 402 
Non-permanent flow, 214 
Non-uniform flow, 270 
Normal pressure. iS 
Nozzles, 132-193, 2$8, 324, 365 
jets from, 135 

Oar, action of, 343 
Obstructions in channels, 243 

in pipes, 172, 1S9 
Ocean waves, 3:8. 350 
Orifices, 40-48, 7i-g7 
Outward-flow turbine. 376, 390, 404 
Overshot wheels, 354, 371 



INDE. 



425 



Paddle wheels, 343, 360 
Parabaloid, 58, 60 
Path of a jet, 61 
Patent log, 336 
Pelton water wheel, 363 
Penstock, 284, 380, 402, 407 
Permanent flow, 214 
Pierce, C. S. ii 

Piezometer, 153, 157, 159, 201, 382 
gaugings, 186, 188 
heights, 192 
Pipes, 162-211, 313, 365 

friction factors for, 168 
friction needs for, 420 
Pitot's tube, 258 
Plates, moving, 317, 338 
Plympton, G. W., 9 
Pneumatic turbine, 400 
Poiseuille's law, 210 
Poncelet wheel, 361 
Power, 2, 283-303 

dynamometer, 291 
Pressure, flow under, 49 

horizontal, 20 

normal, 18 

on planes, 27 

on dams, 28 

on pipes, 18, 167 

transmission of, 14 

unit of, 2 
Pressure gauge, 2, 154, 189 

head, 16, 52, 55, 133, 154 
Probable errors, 91 
Prony brake, 292 
Propeller, 341, 343, 345 
Propulsion, work in, 340 
Pumps, 204, 290 
Pumping, 17, 203 

Range of a jet, 6x, 63, 136 

Rankine, W. J. M., 318 

Reaction, 66, 305 

experiments on, 307 
turbines, 376, 378-410 
wheel, 333, 373 



Rectangular conduits, 224 

orifices, 42, 83 
Reed, E. J., 35 
Relative capacities of pipes, 183 

velocity, 68, 329, 386 
Resistance of plates, 338 

of ships, 336, 337 
Revolving tubes, 332 

vanes, 326, 328 
vessel, 58 
Reynolds, O., 169 
Ring nozzle, 132, 136 
Rivers, 247-282 

curves in, 315 
power of, 302 
Rochester water pipe, 186 
Rod float, 257 
Rossetti, G., 5 
Rounded crests, 117 
Rudder, action of, 348 

RUHLMANN, M., 308, 401 

Salt water, 6 
Screw propeller, 345 

turbine, 401 
Sellers, C, 409 
Sewers, 235 
Ships, 35, 336, 340 
Short pipes, 128, 179 
Siphon, 192 
Slagg, C, 382 
Slope, 213, 244, 248 
Smith, H., Jr., 5, u, 79, 81, 91, 104, 

107, no, 113, 131, 168, 218,226 
Smooth nozzle, 132, 136 
Special forms of turbines, 400 

wheels, 371 
Speed of wheels, 293, 298, 331, 357 

of turbines, 383, 399, 408 
Sphere, 19, 57 
Squares, table of, 416 
Stability of dams, 29 

flotation, 33 
Standard orifice, 71 

tube, 128, 145 



426 



INDEX. 



Standpipe, 206 
Statical moment, 25 
Steamer, coal used by, 340 
Stearns, F. P., 107, 114, 116, 117, ni 

188, 231, 245, 255 
Stevenson, T., 318 
Submerged bodies, 30 
orifices, 86 
surfaces, 18 
turbines, 378 
weirs, 114 
Sub-surface float, 256 

velocities, 245, 264 
Sudbury conduit, 245, 246 
Suppressed weirs, 98, no 
Suppression of contraction, 88 
Surface curve, 122, 273 

velocity, 262 
Surfaces at rest, 310 

centre of pressure, 22, 24 
pressure on, 18, 20 
motion of, 318, 338 
Square orifices, 81 

Tables, see page viii 
Temperature, 6, 8, 211 
Test of a motor, 291, 295 
turbine, 296, 300 
Thearle, S. J. P., 336 
Theoretical hydraulics, 36-70 

discharge, 36 
Theoretic power, 283 
velocity, 40 
Thomson, J., 125 
Thomson, W., 316 
Throttle valve, 171 
Thrust bearing, 408 
Thurston, R. H., 294, 301 
Tidal waves, 349 
Tide gate, 27, 28 
Tides, 303, 344 
Time, 2, 56, 97 
Torricelli, 37 
Transmission of pressures, 14 
Transporting power, 251 



Trapezoidal sections, 227 

weirs, 125 
Triangular orifices, 44 
weirs, 124 
Triple nozzle, 364 
Troughs, 224 
Tubes, 128-161 
Tunnel, Niagara, 403 
Turbines, 295-301, 373-409 
Twin screws, 347 

turbines, 381, 405 



Undershot wheels, 360, 372 
Uniform flow, 214 
Units of measure, 1, 335 
Unwin, W. C, 91, 309, 339 

Vacuum, 8, 10, 17 

in compound tube, 140 

in standard tube, 146 

in turbines, 395 
Valves, 171 
Vanes, 65, 318. 379 

in motion, 318, 322, 325 
revolving, 326, 379 
Variations in discharge, 268 
Velocity, 2, 36, 37 

absolute, 68 

coefficient of, 74 

critical, 210 

from orifices, 37, 73 

in conduits, 217 

in pipes, 173 

in rivers, 249 

of approach, 46, 103, 113 

to move materials, 241, 253 

relative, 6S, 329 
Velocity-head, 39, 40, 52, 106 
Venturi, J. B., 12S, 137, 146 
Venturi water meter, 15S 
Vertical jets, 129, 135 
turbines, 3S1 
wheels, 352, 363 
Very small pipes, 210 






INDEX. 



427 



Waste of water, 198 
weirs, 119 

Water, barometer, 7, 8, 396 
boiling point of, 8 
compressibility, 9 
mains, 198, 201 
maximum density, 5 
measurement of, 72, 90 
meters, 158, 254, 287, 303 
physical properties, 3 
pipes, 162-211 
power, 283-303 
pressure of, 14, 18, 52 
supply systems, 202, 2©6 
surface of, 15, 70 
wastes of, 197 
weight of, 6 



Water, wheels, 289, 352-372 

Waves, 318, 341, 349 

Weight of submerged bodies, 30, 252 

of water, 6 
Weirs, 98-127, 295 
Weisbach, J., 73, 75, 170, 171, 307, 

372, 375 
Wetted perimeter, 212 
Wheel pit, 402 
Wide crests, 118 
Wood, DeV., 58 
Work, denned, 2, 284 

from vanes, 322, 328 

in propulsion, 340 

in pumping, 203 

of friction, 216, 337 

of motors, 283-294, 353 



HYDRAULICS. 



Water-wheels— Windmills — Service Pipe— Drainage, Etc. 

HYDRAULICS AND HYDRAULIC MOTORS. 

WEISBACH. With numerous practical examples for the calculation and 
DU BOIS. construction of Water-wheels, including 1 Breast, Undershot, 
Back -pitch, Overshot Wheels, etc., as well as a special discus- 
sion of the various forms of Turbines. Translated from the 
fourth edition of Weisbach's Mechanics, by A. J. Du Bois. 
Profusely illustrated. Second edition 8vo, cloth, $5 00 



HYDRAULICS. 

SMITH. The Flow of Water through Orifices, over Weirs, and through 

Open Conduits and Pipes. With numerous tables deduced 
from actual experiments. By Hamilton Smith, Jr. 15 folding 

plates. 362 pages 4to, cloth, 

" For the best, most authoritative and most recent work on the flow of 
water through pipe?, we refer our readers to Hamilton Smith's ' Hydrau- 
lics. ,,, —Fire and Water. 



8 00 



A TREATISE ON HYDRAULICS. 

MERRIMAN. Designed as a Text-book for Technical Schools and for the use of 

Engineers. By Prof. Mansfield Merriman, Lehigh University. 

Fourth edition, revised 8vo, clotb, 

"With a tolerably complete knowledge of what has been written on 
Hydraulics in England, Prance, Germany^United States, and to some extent 
Italy, I have no hesitation in saying that I hold this book to be the best 
treatise for students, young or old, yet written. It better presents the 
primary essentials of the art."— From Clemens Herschel, Hydraulic 
Engineer of the Holyoke Water Power Company. 



3 50 



MANUAL OF IRRIGATION ENGINEERING. 

WILSON. By Herbert M.Wilson, C.E. Parti. Hydrography— Includ- 
ing Rainfall, Evaporation and Absorption, Run- off and Flow 
of Streams. Sub-surface Water Sources, Alkali, Drainage and 
Sedimentation, Quantity of Water Required, Pressure and 
Motion of Water, Flow and Measurement of Water in Open 
Canals. Part II. Canals and Canal Works. — Including Classes 
of Irrigation Works, Head works and Diversion Weirs, Scouring 
Sluices, Regulators and Escapes, Falls and Drainage Works 
Distributaries. Part III. Storage Reservoirs. — Including 
Location and Capacity, Earth and Loose Rock Dams, Masonry 
Dams, Waste Ways and Outlet Sluices, Pumping, Tools and 
Maintenance 8vo, cloth, 



4 00 



A GENERAL FORMULA FOR THE UNIFORM 
FLOW OF WATER IN RIVERS AND OTHER 
CHANNELS. 

GANGUILLET By E. Ganguillet and W. R. Kutter, engineers in Berne, 

AND KUTTER. Switzerland. Translated from the German, with numerous 

additions, including Tables, Diagrams, and the elements of 

HERING and 1,200 Gaugings of Rivers, Small Channels, and Pipes, in 

TRAUTWINE. English measure, by Rudolph Hering and John C. Trautwine, 

Jr., of the Society and Institute of Engineers. Illustrated by 

folding plates. Second edition . .8vo, cloth, 

" The distinction of both authors and translators alone is enough to com- 
mend the work. It is made up largely of tables, and for the hydraulic 
engineer the work may be pronounced simply indispensable. The amount 
of labor involved, both in the original work and in this translation, must 
have been very great, and seems fully warranted by the high character of 
the work produced."— Scientific American. 



4 00 



WATER SUPPLY. 

NICHOLS. Considered mainly from a Chemical and Sanitary standpoint. 

By W. Ripley Nichols. With numerous plates. Fourth edition. 

8vo, cloth, 

" We can commend the book to all who desire to have a clear, compact, 
and readable account of the niany interesting questions connected with the 
subject of water supply."— Sanitary Engineer. 



2 50 



WATER-WHEELS OR HYDRAULIC MOTORS. 

BRESSE. Translated from the French Cours de Me'canique Appliquee, 

MAHAN. par M. Bresse, bv Lieut. F. A. Mahan, and revised by Prof. 

D. H. Mahan. 1876 8vo, cloth, $2 50 

THE WINDMILL AS A PRIME MOVER. 

WOLFF. Comprehending everything of value relating to Windmills, 
their Use, Design, Construction, etc. With many fine illus- 
trations. By A. R. Wolff, M.E., Consulting Engineer. Sec- 
ond edition 8vo, cloth, 3 00 

" An excellent practical treatise which fills a .void."— Railroad Gazette. 

" A work of undoubted value."— American Machinist. 

A POPULAR TREATISE ON THE WINDS. 

FERREL. Comprising the General Motions of the Atmosphere, Monsoons, 

Cyclones, Tornadoes, Waterspouts, Hailstones, etc. By Prof. 

William Ferrel. Second edition 8vo, cloth, 4 00 

"The present volume should be read by every student of meteorology, 
because the author, in his attempt at making the work popular, has ex- 
plained at length and with great clearness, many points which in his other 
writings have hot been given sufficiently in detail to make them fully under- 
stood without spending considerable time in thinking over the matter ; and 
also, the views set forth in the work may be considered as the statement of 
the author's present ideas on the topics treated. The volume should be 
read by all general readers desiring information as to the motions of the 
atmosphere, their causes and effects, because they will find in it the opinions 
of the foremost authority on this subject in America, if not in the world. " 

—American Journal of Meteorology. 

COLLECTION OF REPORTS (CONDENSED) AND 
OPINIONS OP CHEMISTS IN REGARD TO THE 
USE OF LEAD PIPE FOR SERVICE PIPE. 

KIRKWOOT). In the Distribution of Water for the Supply of Cities. By 

James P. Kirkwood 8vo, cloth, 1 50 

THE PRACTICE OF THE IMPROVEMENT OF THE 
NON-TIDAL RIVERS OF THE UNITED STATES. 

RUFFNER. With an Examination of the Results thereof. By Capt. E. H. 

Ruffher, Corps of Engineers, U. S. Army 8vo, cloth, 1 25 

"I have no hesitation in saying that it is a work of great value to the 
Corps of Engineers and to others interested in the object of which it treats." 
—Major P. C. Hains, Corps of Engineers, U.S.A. 

A HISTORY OF THE MISSISSIPPI JETTIES AT 
THE MOUTH OF THE MISSISSIPPI RIVER. 

CORTHELL By E. L. Corthell, C.E., Chief Assistant and Resident En- 
gineer during their construction. Illustrated with portrait of 
Captain Eads. 24 page-plates and 10 folding maps. New- 
edition 8vo, cloth, 3 00 

" The book possesses an evident value for public libraries and for scientific 
men. The greatest single undertaking for the benefit of commerce ever 
carried out by the government of the United States.''— Xeiv York Tribvne. 

PROCEEDINGS OF THE DIVISION OF MARINE 
AND NAVAL ENGINEERING AND NAVAL 
ARCHITECTURE 

Of the International Engineering Congress, held at Chicago, 
July 31st to August 5th, 1893. Edited, under the supervision of 
George W. Melville, Engineer-in-Chief, U.S.N., by Walter 
M. McFarland, passed Assistant Engineer, U.S.X. Thoroughly 
illustrated, with half-tone and other plates, of which about 
174 are lull-page, and 126 folding plates. 2 vols., 1400 pages. 

8vo, half morocco, net, 10 00 



Published and for sale by 

JOHN WILEY & SONS, 

53 E. lOth St., New York. 








* V N % A<> " ''/ 









% 




W0U 111 ill 


III 1 1 


1 

'- {'ill in \W# ! i lii 1 lill 
"iniiilm ill fifr \\\ 


■i'irrlfp IP »! 


kfl'ilWifii!jWlM>illril* 



LIBRARY OF CONGRESS 



Il« 

019 971 558 6 






